ContentsArticlesInteger sequence Sequence Integer Abundant
number BaumSweet sequence Bell number Binomial coefficient
Carmichael number Catalan number Composite number Deficient number
Euler numbers Even and odd numbers Factorial Fibonacci number
Fibonacci word Figurate numbers Golomb sequence Happy number Highly
totient number Highly composite number Home prime Hyperperfect
number Juggler sequence Kolakoski sequence Lucky number Lucas
number Padovan sequence Partition number Perfect number
Pseudoperfect number Prime number Pseudoprime Regular paperfolding
sequence 1 3 7 11 12 13 19 35 39 48 49 50 51 54 68 85 88 90 91 96
97 100 101 106 107 109 111 113 117 125 129 130 146 147
RudinShapiro sequence Semiperfect number Semiprime Superperfect
number Thue-Morse sequence Ulam numbers Weird number Recursion
theory Definable set Countable Uncountable Cardinality Beth one
Complete sequence
149 150 151 153 154 158 160 161 171 173 179 181 184 187
ReferencesArticle Sources and Contributors Image Sources,
Licenses and Contributors 189 193
Article LicensesLicense 194
Integer sequence
1
Integer sequenceIn mathematics, an integer sequence is a
sequence (i.e., an ordered list) of integers. An integer sequence
may be specified explicitly by giving a formula for its nth term,
or implicitly by giving a relationship between its terms. For
example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, (the Fibonacci
sequence) is formed by starting with 0 and 1 and then adding any
two consecutive terms to obtain the next one: an implicit
description. The sequence 0, 3, 8, 15, is formed according to the
formula n21 for the nth term: an explicit definition.
Alternatively, an integer sequence may be defined by a property
which members of the sequence possess and other integers do not
possess. For example, we can determine whether a given integer is a
perfect number, even though we do not have a formula for the nth
perfect number.
ExamplesInteger sequences which have received their own name
include: Abundant numbers BaumSweet sequence Bell numbers Binomial
coefficients Carmichael numbers Catalan numbers Composite numbers
Deficient numbers Euler numbers Even and odd numbers Factorial
numbers Fibonacci numbers Fibonacci word Figurate numbers Golomb
sequence Happy numbers Highly totient numbers Highly composite
numbers Home primes Hyperperfect numbers Juggler sequence Kolakoski
sequence Lucky numbers Lucas numbers Padovan numbers Partition
numbers Perfect numbers Pseudoperfect numbers Prime numbers
Pseudoprime numbers Regular paperfolding sequence RudinShapiro
sequence
Integer sequence Semiperfect numbers Semiprime numbers
Superperfect numbers Thue-Morse sequence Ulam numbers Weird
numbers
2
Computable and definable sequencesAn integer sequence is a
computable sequence, if there exists an algorithm which given n,
calculates an, for all n > 0. An integer sequence is a definable
sequence, if there exists some statement P(x) which is true for
that integer sequence x and false for all other integer sequences.
The set of computable integer sequences and definable integer
sequences are both countable, with the computable sequences a
proper subset of the definable sequences (in other words, some
sequences are definable but not computable). The set of all integer
sequences is uncountable (with cardinality equal to that of the
continuum); thus, almost all integer sequences are uncomputable and
cannot be defined.
Complete sequencesAn integer sequence is called a complete
sequence if every positive integer can be expressed as a sum of
values in the sequence, using each value at most once.
External links Journal of Integer Sequences [1]. Articles are
freely available online. Inductive Inference of Integer Sequences
[2]
References[1] http:/ / www. math. uwaterloo. ca/ JIS/ index.
html [2] http:/ / www. cs. cmu. edu/ afs/ cs/ user/ mjs/ ftp/
thesis-program/ 2010/ theses/ tetruashvili. pdf
Sequence
3
SequenceIn mathematics, a sequence is an ordered list of objects
(or events). Like a set, it contains members (also called elements
or terms), and the number of terms (possibly infinite) is called
the length of the sequence. Unlike a set, order matters, and
exactly the same elements can appear multiple times at different
positions in the sequence. A sequence is a discrete function. For
example, (C, R, Y) is a sequence of letters that differs from (Y,
C, R), as the ordering matters. Sequences can be finite, as in this
example, or infinite, such as the sequence of all even positive
integers (2, 4, 6,...). Finite sequences are sometimes known as
strings or words and infinite sequences as streams. The empty
sequence() is included in most notions of sequence, but may be
excluded depending on the context.
Examples and notationThere are various and quite different
notions of sequences in mathematics, some of which (e.g., exact
sequence) are not covered by the notations introduced below. In
addition to identifying the elements of a sequence by their
position, such as "the 3rd element", elements may be given names
for convenient referencing. For example a sequence might be written
as (a1, a2, a2, ), or (b0, b1, b2, ), or (c0, c2, c4, ), depending
on what is useful in the application.
An infinite sequence of real numbers (in blue). This sequence is
neither increasing, nor decreasing, nor convergent, nor Cauchy. It
is bounded, however.
Finite and infiniteA more formal definition of a finite sequence
with terms in a set S is a function from {1, 2, ..., n} to S for
some n 0. An infinite sequence in S is a function from {1, 2, ... }
to S. For example, the sequence of prime numbers (2,3,5,7,11, ) is
the function 12, 23, 35, 47, 511, . A sequence of a finite length n
is also called an n-tuple. Finite sequences include the empty
sequence ( ) that has no elements. A function from all integers
into a set is sometimes called a bi-infinite sequence or two-way
infinite sequence. An example is the bi-infinite sequence of all
even integers ( , -4, -2, 0, 2, 4, 6, 8 ).
Sequence
4
MultiplicativeLet A=(asequence defined by a function f:{1, 2, 3,
...} {1, 2, 3, ...}, such that a i = f(i). The sequence is
multiplicative if f(xy) = f(x)f(y) for all x,y such that x and y
are coprime.[1]
Types and properties of sequencesA subsequence of a given
sequence is a sequence formed from the given sequence by deleting
some of the elements without disturbing the relative positions of
the remaining elements. If the terms of the sequence are a subset
of an ordered set, then a monotonically increasing sequence is one
for which each term is greater than or equal to the term before it;
if each term is strictly greater than the one preceding it, the
sequence is called strictly monotonically increasing. A
monotonically decreasing sequence is defined similarly. Any
sequence fulfilling the monotonicity property is called monotonic
or monotone. This is a special case of the more general notion of
monotonic function. The terms nondecreasing and nonincreasing are
used in order to avoid any possible confusion with strictly
increasing and strictly decreasing, respectively. If the terms of a
sequence are integers, then the sequence is an integer sequence. If
the terms of a sequence are polynomials, then the sequence is a
polynomial sequence. If S is endowed with a topology, then it
becomes possible to consider convergence of an infinite sequence in
S. Such considerations involve the concept of the limit of a
sequence. If A is a set, the free monoid over A (denoted A*) is a
monoid containing all the finite sequences (or strings) of zero or
more elements drawn from A, with the binary operation of
concatenation. The free semigroup A+ is the subsemigroup of A*
containing all elements except the empty sequence.
Sequences in analysisIn analysis, when talking about sequences,
one will generally consider sequences of the form or which is to
say, infinite sequences of elements indexed by natural numbers. It
may be convenient to have the sequence start with an index
different from 1 or 0. For example, the sequence defined by xn =
1/log(n) would be defined only for n 2. When talking about such
infinite sequences, it is usually sufficient (and does not change
much for most considerations) to assume that the members of the
sequence are defined at least for all indices large enough, that
is, greater than some given N. The most elementary type of
sequences are numerical ones, that is, sequences of real or complex
numbers. This type can be generalized to sequences of elements of
some vector space. In analysis, the vector spaces considered are
often function spaces. Even more generally, one can study sequences
with elements in some topological space.
Sequence
5
SeriesThe sum of terms of a sequence is a series. More
precisely, if (x1, x2, x3, ...) is a sequence, one may consider the
sequence of partial sums (S1, S2, S3, ...), with
Formally, this pair of sequences comprises the series with the
terms x1, x2, x3, ..., which is denoted as
If the sequence of partial sums is convergent, one also uses the
infinite sum notation for its limit. For more details, see
series.
Infinite sequences in theoretical computer scienceInfinite
sequences of digits (or characters) drawn from a finite alphabet
are of particular interest in theoretical computer science. They
are often referred to simply as sequences or streams, as opposed to
finite strings. Infinite binary sequences, for instance, are
infinite sequences of bits (characters drawn from the alphabet
{0,1}). The set C = {0, 1} of all infinite, binary sequences is
sometimes called the Cantor space. An infinite binary sequence can
represent a formal language (a set of strings) by setting the n th
bit of the sequence to 1 if and only if the n th string (in
shortlex order) is in the language. Therefore, the study of
complexity classes, which are sets of languages, may be regarded as
studying sets of infinite sequences. An infinite sequence drawn
from the alphabet {0, 1, ..., b1} may also represent a real number
expressed in the base-b positional number system. This equivalence
is often used to bring the techniques of real analysis to bear on
complexity classes.
Sequences as vectorsSequences over a field may also be viewed as
vectors in a vector space. Specifically, the set of F-valued
sequences (where F is a field) is a function space (in fact, a
product space) of F-valued functions over the set of natural
numbers. In particular, the term sequence space usually refers to a
linear subspace of the set of all possible infinite sequences with
elements in .
Doubly-infinite sequencesNormally, the term infinite sequence
refers to a sequence which is infinite in one direction, and finite
in the otherthe sequence has a first element, but no final element
(a singly-infinite sequence). A doubly-infinite sequence is
infinite in both directionsit has neither a first nor a final
element. Singly-infinite sequences are functions from the natural
numbers (N) to some set, whereas doubly-infinite sequences are
functions from the integers (Z) to some set. One can interpret
singly infinite sequences as elements of the semigroup ring of the
natural numbers doubly infinite sequences as elements of the group
ring of the integers Cauchy product of sequences. , and
. This perspective is used in the
Sequence
6
Ordinal-indexed sequenceAn ordinal-indexed sequence is a
generalization of a sequence. If is a limit ordinal and X is a set,
an -indexed sequence of elements of X is a function from to X. In
this terminology an -indexed sequence is an ordinary sequence.
Sequences and automataAutomata or finite state machines can
typically be thought of as directed graphs, with edges labeled
using some specific alphabet . Most familiar types of automata
transition from state to state by reading input letters from ,
following edges with matching labels; the ordered input for such an
automaton forms a sequence called a word (or input word). The
sequence of states encountered by the automaton when processing a
word is called a run. A nondeterministic automaton may have
unlabeled or duplicate out-edges for any state, giving more than
one successor for some input letter. This is typically thought of
as producing multiple possible runs for a given word, each being a
sequence of single states, rather than producing a single run that
is a sequence of sets of states; however, 'run' is occasionally
used to mean the latter.
Types of sequences 1-sequence Arithmetic progression Cauchy
sequence Farey sequence Fibonacci sequence Geometric progression
Look-and-say sequence ThueMorse sequence
Related concepts List (computing) Ordinal-indexed sequence
Recursion (computer science) Tuple Set theory
Operations on sequences Cauchy product Limit of a sequence
References[1] Lando, Sergei K.. "7.4 Multiplicative sequences".
Lectures on generating functions. AMS. ISBN0821834819.
External links The On-Line Encyclopedia of Integer Sequences
(http://www.research.att.com/~njas/sequences/index.html) Journal of
Integer Sequences
(http://www.cs.uwaterloo.ca/journals/JIS/index.html) (free)
Sequence
(http://planetmath.org/?op=getobj&from=objects&id=397) on
PlanetMath
Integer
7
IntegerThe integers (from the Latin integer, literally
"untouched", hence "whole": the word entire comes from the same
origin, but via French[1] ) are formed by the natural numbers
including 0 (0, 1, 2, 3, ...) together with the negatives of the
non-zero natural numbers (1, 2, 3, ...). Viewed as a subset of the
real numbers, they are numbers that can be written without a
fractional or decimal component, and fall within the set {... 2, 1,
0, 1, 2, ...}. For example, 65, 7, and 759 are integers; 1.6 and 1
are not integers.Symbol often used to denote the set of
integers
The set of all integers is often denoted by a boldface Z (or
blackboard bold for Zahlen (German for numbers, pronounced integers
modulo n (for example, ).German pronunciation:[tsaln]).
, Unicode U+2124 ), which stands[2]
The set
is the finite set of
The integers (with addition as operation) form the smallest
group containing the additive monoid of the natural numbers. Like
the natural numbers, the integers form a countably infinite set. In
algebraic number theory, these commonly understood integers,
embedded in the field Integers can be thought of as discrete,
equally spaced points on an infinitely long of rational numbers,
are referred to as number line. rational integers to distinguish
them from the more broadly defined algebraic integers (but with
"rational" meaning "quotient of integers", this attempt at
precision suffers from circularity).
Algebraic propertiesLike the natural numbers, Z is closed under
the operations of addition and multiplication, that is, the sum and
product of any two integers is an integer. However, with the
inclusion of the negative natural numbers, and, importantly, zero,
Z (unlike the natural numbers) is also closed under subtraction. Z
is not closed under division, since the quotient of two integers
(e.g., 1 divided by 2), need not be an integer. Although the
natural numbers are closed under exponentiation, the integers are
not (since the result can be a fraction when the exponent is
negative). The following lists some of the basic properties of
addition and multiplication for any integers a, b and c.Addition
Closure: Associativity: Commutativity: a+b is an integer
Multiplication ab is an integer
a+(b+c)=(a+b)+c a(bc)=(ab)c a+b=b+a ab=ba a1=a An inverse
element usually does not exist at all.
Existence of an identity element: a+0=a Existence of inverse
elements: Distributivity: No zero divisors: a+(a)=0
a (b + c) = (a b) + (a c) and (a + b) c = (a c) + (b c) If ab =
0, then a = 0 or b = 0 (or both)
In the language of abstract algebra, the first five properties
listed above for addition say that Z under addition is an abelian
group. As a group under addition, Z is a cyclic group, since every
nonzero integer can be written as a finite sum 1 + 1 + ... + 1 or
(1) + (1) + ... + (1). In fact, Z under addition is the only
infinite cyclic group, in the sense
Integer that any infinite cyclic group is isomorphic to Z. The
first four properties listed above for multiplication say that Z
under multiplication is a commutative monoid. However not every
integer has a multiplicative inverse; e.g. there is no integer x
such that 2x = 1, because the left hand side is even, while the
right hand side is odd. This means that Z under multiplication is
not a group. All the rules from the above property table, except
for the last, taken together say that Z together with addition and
multiplication is a commutative ring with unity. Adding the last
property says that Z is an integral domain. In fact, Z provides the
motivation for defining such a structure. The lack of
multiplicative inverses, which is equivalent to the fact that Z is
not closed under division, means that Z is not a field. The
smallest field containing the integers is the field of rational
numbers. This process can be mimicked to form the field of
fractions of any integral domain. Although ordinary division is not
defined on Z, it does possess an important property called the
division algorithm: that is, given two integers a and b with b0,
there exist unique integers q and r such that a = q b + r and 0 r
< | b |, where | b | denotes the absolute value of b. The
integer q is called the quotient and r is called the remainder,
resulting from division of a by b. This is the basis for the
Euclidean algorithm for computing greatest common divisors. Again,
in the language of abstract algebra, the above says that Z is a
Euclidean domain. This implies that Z is a principal ideal domain
and any positive integer can be written as the products of primes
in an essentially unique way. This is the fundamental theorem of
arithmetic.
8
Order-theoretic propertiesZ is a totally ordered set without
upper or lower bound. The ordering of Z is given by: ... 3 < 2
< 1 < 0 < 1 < 2 < 3 < ... An integer is positive
if it is greater than zero and negative if it is less than zero.
Zero is defined as neither negative nor positive. The ordering of
integers is compatible with the algebraic operations in the
following way: 1. if a < b and c < d, then a + c < b + d
2. if a < b and 0 < c, then ac < bc. It follows that Z
together with the above ordering is an ordered ring. The integers
are the only integral domain whose positive elements are
well-ordered, and in which order is preserved by addition.
Integer
9
ConstructionThe integers can be formally constructed as the
equivalence classes of ordered pairs of natural numbers (a, b). The
intuition is that (a, b) stands for the result of subtracting b
from a. To confirm our expectation that 1 2 and 4 5 denote the same
number, we define an equivalence relation ~ on these pairs with the
following rule:
alt=Representation of equivalence classes for the numbers -5 to
5 |Red Points represent ordered pairs of natural numbers. Linked
red points are equivalence classes representing the blue integers
at the end of the line. |upright=2
precisely when
Addition and multiplication of integers can be defined in terms
of the equivalent operations on the natural numbers; denoting by
[(a,b)] the equivalence class having (a,b) as a member, one
has:
The negation (or additive inverse) of an integer is obtained by
reversing the order of the pair:
Hence subtraction can be defined as the addition of the additive
inverse:
The standard ordering on the integers is given by: iff It is
easily verified that these definitions are independent of the
choice of representatives of the equivalence classes. Every
equivalence class has a unique member that is of the form (n,0) or
(0,n) (or both at once). The natural number n is identified with
the class [(n,0)] (in other words the natural numbers are embedded
into the integers by map sending n to [(n,0)]), and the class
[(0,n)] is denoted n (this covers all remaining classes, and gives
the class [(0,0)] a second time since 0=0. Thus, [(a,b)] is denoted
by
If the natural numbers are identified with the corresponding
integers (using the embedding mentioned above), this convention
creates no ambiguity. This notation recovers the familiar
representation of the integers as {... 3,2,1, 0, 1, 2, 3, ...}.
Some examples are:
Integer
10
Integers in computingAn integer is often a primitive datatype in
computer languages. However, integer datatypes can only represent a
subset of all integers, since practical computers are of finite
capacity. Also, in the common two's complement representation, the
inherent definition of sign distinguishes between "negative" and
"non-negative" rather than "negative, positive, and 0". (It is,
however, certainly possible for a computer to determine whether an
integer value is truly positive.) Fixed length integer
approximation datatypes (or subsets) are denoted int or Integer in
several programming languages (such as Algol68, C, Java, Delphi,
etc.). Variable-length representations of integers, such as
bignums, can store any integer that fits in the computer's memory.
Other integer datatypes are implemented with a fixed size, usually
a number of bits which is a power of 2 (4, 8, 16, etc.) or a
memorable number of decimal digits (e.g., 9 or 10).
CardinalityThe cardinality of the set of integers is equal to If
N = {0, 1, 2, ...} then consider the function: (aleph-null). This
is readily demonstrated by the construction of a bijection, that
is, a function that is injective and surjective from Z to N.
{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,0) (1,1) (2,3) (3,5) ... }
If N = {1,2,3,...} then consider the function:
{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,1) (1,3) (2,5) (3,7) ... }
If the domain is restricted to Z then each and every member of Z
has one and only one corresponding member of N and by the
definition of cardinal equality the two sets have equal
cardinality.
Integer
11
Notes[1] Evans, Nick (1995). "A-Quantifiers and Scope" (http:/ /
books. google. com/ ?id=NlQL97qBSZkC). In Bach, Emmon W.
Quantification in Natural Languages. Dordrecht, The Netherlands;
Boston, MA: Kluwer Academic Publishers. pp.262. ISBN0792333527. [2]
Miller, Jeff (2010-08-29). "Earliest Uses of Symbols of Number
Theory" (http:/ / jeff560. tripod. com/ nth. html). . Retrieved
2010-09-20.
References Bell, E. T., Men of Mathematics. New York: Simon and
Schuster, 1986. (Hardcover; ISBN 0-671-46400-0)/(Paperback; ISBN
0-671-62818-6) Herstein, I. N., Topics in Algebra, Wiley; 2 edition
(June 20, 1975), ISBN 0-471-01090-1. Mac Lane, Saunders, and
Garrett Birkhoff; Algebra, American Mathematical Society; 3rd
edition (April 1999). ISBN 0-8218-1646-2. Weisstein, Eric W., "
Integer (http://mathworld.wolfram.com/Integer.html)" from
MathWorld.
External links The Positive Integers - divisor tables and
numeral representation tools (http://www.positiveintegers.org)
On-Line Encyclopedia of Integer Sequences
(http://www.research.att.com/~njas/sequences/) cf OEIS This article
incorporates material from Integer on PlanetMath, which is licensed
under the Creative Commons Attribution/Share-Alike License.
Abundant numberIn number theory, an abundant number or excessive
number is a number n for which (n) > 2n. Here (n) is the
sum-of-divisors function: the sum of all positive divisors of n,
including n itself. The value (n)2n is called the abundance of n.
An equivalent definition is that the proper divisors of the number
(the divisors except the number itself) sum to more than the
number. The first few abundant numbers (sequence A005101 [1] in
OEIS) are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70,
72, 78, 80, 84, 88, 90, 96, 100, 102, As an example, consider the
number 24. Its divisors are 1,2, 3, 4, 6, 8, 12 and 24, whose sum
is 60. Because 60 is more than 224, the number 24 is abundant. Its
abundance is 60224=12. The smallest abundant number not divisible
by two, i.e. odd, is 945, and the smallest not divisible by 2 or by
3 is 5391411025 whose prime factors are 52, 7, 11, 13, 17, 19, 23,
and 29. An algorithm given by Iannucci in 2005 shows how to find
the smallest abundant number not divisible by the first k
primes.[2] If represents the smallest abundant number not divisible
by the first k primes then for all we have: for k sufficiently
large. Infinitely many even and odd abundant numbers exist. Marc
Delglise showed in 1998 that the natural density of the set of
abundant numbers and perfect numbers is between 0.2474 and
0.2480.[3] Every proper multiple of a perfect number, and every
multiple of an abundant number, is abundant. Also, every integer
greater than 20161 can be written as the sum of two abundant
numbers.[4] An abundant number which is not a semiperfect number is
called a weird number; an abundant number with abundance 1 is
called a quasiperfect number. Closely related to abundant numbers
are perfect numbers with (n)=2n, and deficient numbers with (n)
stop if stop < start: start, stop = stop, start if start < 1:
start = 1 if stop < 1: stop = 1 t = [[1]] two-dimensional array
c = 1 while c = start: yield t[-1][0] previous row row =
[t[-1][-1]] for b in t[-1]: row.append(row[-1] + b) c += 1 number
t.append(row) for b in bell_numbers(1, 300): print b The number in
the nth row and kth column is the number of partitions of {1, ...,
n} such that n is not together in one class with any of the
elements k, k+1, ..., n1. For example, there are 7 partitions of
{1, ..., 4} such that 4 is not together in one class with either of
the elements 2, 3, and there are 10 partitions of {1, ..., 4} such
that 4 is not together in one class with element 3. The difference
is due to 3 partitions of {1, ..., 4} such that 4 is together in
one class with element 2, but not with element 3. This corresponds
to the fact that there are 3 partitions of {1, ..., 3} such that 3
is not together in one class with element 2: for counting
partitions two elements which are always in one class can be
treated as just one element. The 3 appears in the previous row of
the table. ## Initialize the triangle as a ## Bell numbers
count
18
## Yield the Bell number of the ## Initialize a new row ##
Populate the new row ## We have found another Bell ## Append the
row to the triangle
Prime Bell numbersThe first few Bell numbers that are primes
are: 2, 5, 877, 27644437, 35742549198872617291353508656626642567,
359334085968622831041960188598043661065388726959079837
corresponding to the indices 2, 3, 7, 13, 42 and 55 (sequence
A051130 [3] in OEIS). The next prime is B2841, which is
approximately 9.30740105 106538. [4] As of 2006, it is the largest
known prime Bell number. Phil Carmody showed it was a probable
prime in 2002. After 17 months of computation with Marcel Martin's
ECPP program Primo, Ignacio Larrosa Caestro proved it to be prime
in 2004. He ruled out any other possible primes below B6000, later
extended to B30447 by Eric Weisstein.
Bell number
19
References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000110
[2] Spivey, Michael (2008), "A Generalized Recurrence for Bell
Numbers" (http:/ / www. cs. uwaterloo. ca/ journals/ JIS/ VOL11/
Spivey/ spivey25. pdf), Journal of Integer Sequences 11, [3] http:/
/ en. wikipedia. org/ wiki/ Oeis%3Aa051130 [4] http:/ / primes.
utm. edu/ primes/ page. php?id=68825
Gian-Carlo Rota (1964). "The Number of Partitions of a Set".
American Mathematical Monthly 71 (5): 498504. doi:10.2307/2312585.
MR0161805. Lovsz, L. (1993). Combinatorial Problems and Exercises
(2nd ed. ed.). Amsterdam, Netherlands: North-Holland. Berend, D.;
Tassa, T. (2010). "Improved Bounds on Bell Numbers and on Moments
of Sums of Random Variables". Probability and Mathematical
Statistics (http://www.math.uni.wroc.pl/~pms/index.php) 30 (2):
185205.
External links Robert Dickau. "Diagrams of Bell numbers"
(http://mathforum.org/advanced/robertd/bell.html). Pat Ballew.
"Bell numbers" (http://www.pballew.net/Bellno.html). Weisstein,
Eric W., " Bell Number
(http://mathworld.wolfram.com/BellNumber.html)" from MathWorld.
Wagstaff, Samuel S. (1996). "Aurifeuillian factorizations and the
period of the Bell numbers modulo a prime"
(http://homes.cerias.purdue.edu/~ssw/bell/bell.ps). Mathematics of
computation 65 (213): 383391. doi:10.1090/S0025-5718-96-00683-7.
MR1325876 Bibcode:1996MaCom..65..383W. Gottfried Helms. "Further
properties & Generalization of Bell-Numbers"
(http://go.helms-net.de/math/
binomial/04_5_SummingBellStirling.pdf).
Binomial coefficientIn mathematics, binomial coefficients are a
family of positive integers that occur as coefficients in the
binomial theorem. They are indexed by two nonnegative integers; the
binomial coefficient indexed by n and k is usually written , and it
is the coefficient of the xk term in the polynomial expansion of
the binomial power (1+x)n. Arranging binomial coefficients into
rows for successive values of n, and in which k ranges from 0 to n,
gives a triangular array called Pascal's triangle. This family of
numbers also arises in many other areas than The binomial
coefficients can be arranged to form algebra, notably in
combinatorics. For any set containing n Pascal's triangle.
elements, the number of distinct k-element subsets of it that can
be formed (the k-combinations of its elements) is given by the
binomial coefficient . Therefore is often read as "n choose k". The
properties of binomial coefficients have led to extending the
meaning of the symbol beyond the basic case where n and k are
nonnegative integers with k
n; such expressions are then still called binomial coefficients.
The notation was introduced by Andreas von Ettingshausen in
1826,[1] although the numbers were already
known centuries before that (see Pascal's triangle). The
earliest known detailed discussion of binomial coefficients is in a
tenth-century commentary, due to Halayudha, on an ancient Hindu
classic, Pingala's chandastra. In about 1150, the Hindu
mathematician Bhaskaracharya gave a very clear exposition of
binomial coefficients in his book
Binomial coefficient Lilavati.[2] Alternative notations include
C(n, k), nCk, nCk, , ,[3] in all of which the C stands for
combinations or choices.
20
Definition and interpretationsFor natural numbers (taken to
include 0) n and k, the binomial coefficient can be defined as the
coefficient of the monomial Xk in the expansion of (1 + X)n. The
same coefficient also occurs (if k n) in the binomial formula
(valid for any elements x,y of a commutative ring), which
explains the name "binomial coefficient". Another occurrence of
this number is in combinatorics, where it gives the number of ways,
disregarding order, that k objects can be chosen from among n
objects; more formally, the number of k-element subsets (or
k-combinations) of an n-element set. This number can be seen to be
equal to the one of the first definition, independently of any of
the formulas below to compute it: if in each of the n factors of
the power (1 + X)n one temporarily labels the term X with an index
i (running from 1 to n), then each subset of k indices gives after
expansion a contribution Xk, and the coefficient of that monomial
in the result will be the number of such subsets. This shows in
particular that is a natural number for any natural numbers n and
k. There are many other combinatorial interpretations of binomial
coefficients (counting problems for which the answer is given by a
binomial coefficient expression), for instance the number of words
formed of n bits (digits 0 or 1) whose sum is k is given by , while
the number of ways to write where every ai is a nonnegative integer
is given by interpretations are easily seen to be equivalent to
counting k-combinations. . Most of these
Computing the value of binomial coefficientsSeveral methods
exist to compute the value of k-combinations. without actually
expanding a binomial power or counting
Recursive formulaOne has a recursive formula for binomial
coefficients
with initial values
The formula follows either from tracing the contributions to Xk
in (1 + X)n1(1 + X), or by counting k-combinations of {1, 2, ...,
n} that contain n and that do not contain n separately. It follows
easily that when k>n, and for all n, so the recursion can stop
when reaching such cases. This recursive formula then allows the
construction of Pascal's triangle.
Binomial coefficient
21
Multiplicative formulaA more efficient method to compute
individual binomial coefficients is given by the formula
This formula is easiest to understand for the combinatorial
interpretation of binomial coefficients. The numerator gives the
number of ways to select a sequence of k distinct objects,
retaining the order of selection, from a set of n objects. The
denominator counts the number of distinct sequences that define the
same k-combination when order is disregarded.
Factorial formulaFinally there is a formula using factorials
that is easy to remember:
where n! denotes the factorial of n. This formula follows from
the multiplicative formula above by multiplying numerator and
denominator by (n k)!; as a consequence it involves many factors
common to numerator and denominator. It is less practical for
explicit computation unless common factors are first canceled (in
particular since factorial values grow very rapidly). The formula
does exhibit a symmetry that is less evident from the
multiplicative formula (though it is from the definitions)
Generalization and connection to the binomial seriesThe
multiplicative formula allows the definition of binomial
coefficients to be extended[4] by replacing n by an arbitrary
number (negative, real, complex) or even an element of any
commutative ring in which all positive integers are invertible:
With this definition one has a generalization of the binomial
formula (with one of the variables set to 1), which justifies still
calling the binomial coefficients:
This formula is valid for all complex numbers and X with |X|n
are zero, and the infinite series becomes a finite sum, thereby
recovering the binomial formula. However for other values of ,
including negative integers and rational numbers, the series is
really infinite.
Binomial coefficient
22
Pascal's trianglePascal's rule is the important recurrence
relation
which can be used to prove by mathematical induction that
is a natural number for all n and k, (equivalent to the
statement that k! divides the product of k consecutive
integers), a fact that is not immediately obvious from formula (1).
Pascal's rule also gives rise to Pascal's triangle:0: 1: 2: 3: 4:
5: 6: 7: 8: 1 1 8 1 7 28 1 6 21 56 1 5 15 35 70 1 4 10 20 35 56 1 3
6 10 15 21 28 1 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1 1
Row number n contains the numbers
for k = 0,,n. It is constructed by starting with ones at the
outside and
then always adding two adjacent numbers and writing the sum
directly underneath. This method allows the quick calculation of
binomial coefficients without the need for fractions or
multiplications. For instance, by looking at row number 5 of the
triangle, one can quickly read off that (x + y)5 = 1 x5 + 5 x4y +
10 x3y2 + 10 x2y3 + 5 x y4 + 1 y5. The differences between elements
on other diagonals are the elements in the previous diagonal, as a
consequence of the recurrence relation (3) above.
Combinatorics and statisticsBinomial coefficients are of
importance in combinatorics, because they provide ready formulas
for certain frequent counting problems: There are There are There
are There are ways to choose k elements from a set of n elements.
See Combination. ways to choose k elements from a set of n if
repetitions are allowed. See Multiset. strings containing k ones
and n zeros. strings consisting of k ones and n zeros such that no
two ones are adjacent.
The Catalan numbers are The binomial distribution in statistics
is The formula for a Bzier curve.
Binomial coefficient
23
Binomial coefficients as polynomialsFor any nonnegative integer
k, the expression can be simplified and defined as a polynomial
divided by k!:
This presents a polynomial in t with rational coefficients. As
such, it can be evaluated at any real or complex number t to define
binomial coefficients with such first arguments. These "generalized
binomial coefficients" appear in Newton's generalized binomial
theorem. For each k, the polynomial can be characterized as the
unique degree k polynomial p(t) satisfying p(0) = p(1) =
... = p(k 1) = 0 and p(k) = 1. Its coefficients are expressible
in terms of Stirling numbers of the first kind, by definition of
the latter:
The derivative of
can be calculated by logarithmic differentiation:
Binomial coefficients as a basis for the space of
polynomialsOver any field containing Q, each polynomial p(t) of
degree at most d is uniquely expressible as a linear combination .
The coefficient ak is the kth difference of the sequence p(0),
p(1), , p(k). Explicitly,[5]
Integer-valued polynomialsEach polynomial is integer-valued: it
takes integer values at integer inputs. (One way to prove this is
by induction on k, using Pascal's identity.) Therefore any integer
linear combination of binomial coefficient polynomials is
integer-valued too. Conversely, (3.5) shows that any integer-valued
polynomial is an integer linear combination of these binomial
coefficient polynomials. More generally, for any subring R of a
characteristic 0 field K, a polynomial in K[t] takes values in R at
all integers if and only if it is an R-linear combination of
binomial coefficient polynomials.
ExampleThe integer-valued polynomial 3t(3t+1)/2 can be rewritten
as
Identities involving binomial coefficientsFor any nonnegative
integers n and k, This follows from (2) by using (1+x)n =
xn(1+x1)n. It is reflected in the symmetry of Pascal's triangle. A
combinatorial interpretation of this formula is as follows: when
forming a subset of elements (from a set of size ), it is
equivalent to consider the number of ways you can pick elements and
the number of ways you can
Binomial coefficient exclude elements.
24
The factorial definition lets one relate nearby binomial
coefficients. For instance, if k is a positive integer and n is
arbitrary, then
and, with a little more work,
Powers of -1A special binomial coefficient is ; it equals powers
of -1:
Series involving binomial coefficientsThe formula
is obtained from (2) using x = 1. This is equivalent to saying
that the elements in one row of Pascal's triangle always add up to
two raised to an integer power. A combinatorial interpretation of
this fact involving double counting is given by counting subsets of
size 0, size 1, size 2, and so on up to size n of a set S of n
elements. Since we count the number of subsets of size i for 0 i n,
this sum must be equal to the number of subsets of S, which is
known to be 2n. That is, Equation5 is a statement that the power
set for a finite set with n elements has size 2n. The formulas
and
follow from (2), after differentiating with respect to x (twice
in the latter) and then substituting x = 1. The Chu-Vandermonde
identity, which holds for any complex-values m and n and any
non-negative integer k, is
and can be found by examination of the coefficient of
in the expansion of (1+x)m(1+x)nm = (1+x)n using
equation (2). When m=1, equation (7a) reduces to equation (3). A
similar looking formula, which applies for any integers j, k, and n
satisfying 0jkn, is
and
can
be
found
by
examination using
of
the
coefficient
of
in
the
expansion
of
When j=k, equation (7b) gives
Binomial coefficient
25
From expansion (7a) using n=2m, k = m, and (4), one finds
Let F(n) denote the nth Fibonacci number. We obtain a formula
about the diagonals of Pascal's triangle
This can be proved by induction using (3) or by Zeckendorf's
representation (Just note that the lhs gives the number of subsets
of {F(2),...,F(n)} without consecutive members, which also form all
the numbers below F(n+1)). Also using (3) and induction, one can
show that
Again by (3) and induction, one can show that for k = 0, ... ,
n1
as well as
which is itself a special case of the result from the theory of
finite differences that for any polynomial P(x) of degree less than
n,
Differentiating (2) k times and setting x = 1 yields this for
and the general case follows by taking linear combinations of
these. When P(x) is of degree less than or equal to n,
, when 0 k < n,
where
is the coefficient of degree n in P(x).
More generally for 13b,
where m and d are complex numbers. This follows immediately
applying (13b) to the polynomial Q(x):=P(m + dx) instead of P(x),
and observing that Q(x) has still degree less than or equal to n,
and that its coefficient of degree n is dnan. The infinite
series
Binomial coefficient is convergent for k 2. This formula is used
in the analysis of the German tank problem. It is equivalent to the
formula for the finite sum
26
which is proved for M>m by induction on M. Using (8) one can
derive
and .
Identities with combinatorial proofsMany identities involving
binomial coefficients can be proved by combinatorial means. For
example, the following identity for nonnegative integers (which
reduces to (6) when ):
can be given a double counting proof as follows. The left side
counts the number of ways of selecting a subset of of at least q
elements, and marking q elements among those selected. The right
side counts the same parameter, because there are ways of choosing
a set of q marks and they occur in all subsets that additionally
contain some
subset of the remaining elements, of which there are The
recursion formula
where both sides count the number of k-element subsets of {1, 2,
. . . , n} with the right hand side rst grouping them into those
which contain element n and those which do not. The identity (8)
also has a combinatorial proof. The identity reads
Suppose you have
empty squares arranged in a row and you want to mark (select) n
of them. There are
ways to do this. On the other hand, you may select your n
squares by selecting k squares from among the first n and squares
from the remaining n squares. This gives
Now apply (4) to get the result.
Binomial coefficient Sum of coefficients row The number of
k-combinations for all k, , is the sum of the nth row (counting
from 0) of the
27
binomial coefficients. These combinations are enumerated by the
1 digits of the set of base 2 numbers counting from 0 to , where
each digit position is an item from the set of n.
Continuous identitiesCertain trigonometric integrals have values
expressible in terms of binomial coefficients: For and
These can be proved by using Euler's formula to convert
trigonometric functions to complex exponentials, expanding using
the binomial theorem, and integrating term by term.
Generating functionsOrdinary generating functionsFor a fixed n,
the ordinary generating function of the sequence is:
For a fixed k, the ordinary generating function of the
sequence
is:
The bivariate generating function of the binomial coefficients
is:
Another bivariate generating function of the binomial
coefficients, which is symmetric, is:
Binomial coefficient
28
Exponential generating functionThe exponential bivariate
generating function of the binomial coefficients is:
Divisibility propertiesIn 1852, Kummer proved that if m and n
are nonnegative integers and p is a prime number, then the largest
power of p dividing equals pc, where c is the number of carries
when m and n are added in base p. Equivalently, the exponent of a
prime p in equals the number of nonnegative integers j such that
the fractional part of k/pj is greater than the fractional part of
n/pj. It can be deduced from this that is divisible by n/gcd(n,k).
A somewhat surprising result by David Singmaster (1974) is that any
integer divides almost all binomial coefficients. More precisely,
fix an integer d and let f(N) denote the number of binomial
coefficients with n < N such that d divides . Then
Since the number of binomial coefficients
with n < N is N(N+1) / 2, this implies that the density of
binomial
coefficients divisible by d goes to 1. Another fact: An integer
n2 is prime if and only if all the intermediate binomial
coefficients
are divisible by n. Proof: When p is prime, p divides for all 0
1) and (n not in s): s.add(n) n = sum(SQUARE[d] for d in str(n))
return (n == 1)
References[1] [2] [3] [4] [5] [6] "Sad Number" (http:/ /
mathworld. wolfram. com/ SadNumber. html). Wolfram Research, Inc..
. Retrieved 2009-09-16. http:/ / en. wikipedia. org/ wiki/
Oeis%3Aa007770 http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa035497
The Prime Database: 10^150006+7426247*10^75000+1 (http:/ / primes.
utm. edu/ primes/ page. php?id=76550) Prime Pages entry for
242643801 - 1 (http:/ / primes. utm. edu/ primes/ page.
php?id=88847) A161872 (http:/ / en. wikipedia. org/ wiki/
Oeis:a161872)
Additional resources Walter Schneider, Mathews: Happy Numbers
(http://web.archive.org/web/20060204094653/http://www.
wschnei.de/digit-related-numbers/happy-numbers.html). Weisstein,
Eric W., " Happy Number
(http://mathworld.wolfram.com/HappyNumber.html)" from MathWorld.
Happy Numbers (http://mathforum.org/library/drmath/view/55856.html)
at The Math Forum. Guy, Richard (2004). Unsolved Problems in Number
Theory (third edition). Springer-Verlag. ISBN 0-387-20860-7.
External links Reg Allenby page
(http://www.maths.leeds.ac.uk/pure/staff/allenby/allenby.html)
Highly totient number
96
Highly totient numberA highly totient number k is an integer
that has more solutions to the equation (x) = k, where is Euler's
totient function, than any integer below it. The first few highly
totient numbers are 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480,
576, 720, 1152, 1440 (sequence A097942 [1] in OEIS). with 1, 3, 4,
5, 6, 10, 11, 17, 21, 31, 34, 37, 38, 49, 54, and 72 totient
solutions respectively. The sequence of highly totient numbers is a
subset of the sequence of smallest number k with exactly n
solutions to (x) = k. These numbers have more ways of being
expressed as products of numbers of the form p - 1 and their
products than smaller integers. The concept is somewhat analogous
to that of highly composite numbers, and in the same way that 1 is
the only odd highly composite number, it is also the only odd
highly totient number (indeed, the only odd number to not be a
nontotient). And just as there are infinitely many highly composite
numbers, there are also infinitely many highly totient numbers,
though the highly totient numbers get tougher to find the higher
one goes, since calculating the totient function involves
factorization into primes, something that becomes extremely
difficult as the numbers get larger.
References L. Havelock, A Few Observations on Totient and
Cototient Valence [2] from PlanetMath
References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa097942
[2] http:/ / aux. planetmath. org/ files/ papers/ 335/
C:TempObsTotientCototientValence. pdf
Highly composite number
97
Highly composite numberA highly composite number (HCN) is a
positive integer with more divisors than any positive integer
smaller than itself. The initial or smallest twenty-one highly
composite numbers are listed in the table at right.number of
divisors of 1 2 4 6 12 24 36 48 60 120 180 240 360 720 840 1,260
1,680 2,520 5,040 7,560 10,080 1 2 3 4 6 8 9 10 12 16 18 20 24 30
32 36 40 48 60 64 72
The sequence of highly composite numbers (sequence A002182 [1]
in OEIS) is a subset of the sequence of smallest numbers k with
exactly n divisors (sequence A005179 [2] in OEIS). There are an
infinite number of highly composite numbers. To prove this fact,
suppose that n is an arbitrary highly composite number. Then 2n has
more divisors than n (2n itself is a divisor and so are all the
divisors of n) and so some number larger than n (and not larger
than 2n) must be highly composite as well. Roughly speaking, for a
number to be highly composite it has to have prime factors as small
as possible, but not too many of the same. If we decompose a number
n in prime factors like this:
where n is exactly
are prime, and the exponents
are positive integers, then the number of divisors of
Hence, for n to be a highly composite number,
Highly composite number the k given prime numbers pi must be
precisely the first k prime numbers (2, 3, 5, ...); if not, we
could replace one of the given primes by a smaller prime, and thus
obtain a smaller number than n with the same number of divisors
(for instance 10= 25 may be replaced with 6 = 2 3; both have 4
divisors); the sequence of exponents must be non-increasing, that
is ; otherwise, by exchanging two exponents we would again get a
smaller number than n with the same number of divisors (for
instance 18=2132 may be replaced with 12=2231; both have six
divisors). Also, except in two special cases n=4 and n=36, the last
exponent ck must equal1. Saying that the sequence of exponents is
non-increasing is equivalent to saying that a highly composite
number is a product of primorials. Because the prime factorization
of a highly composite number uses all of the first k primes, every
highly composite number must be a practical number.[3] Highly
composite numbers higher than 6 are also abundant numbers. One need
only look at the three or four highest divisors of a particular
highly composite number to ascertain this fact. It is false that
all highly composite numbers are also Harshad numbers in base 10.
The first HCN that is not a Harshad number is 245,044,800, which
has a digit sum of 27, but 27 does not divide evenly into
245,044,800. Many of these numbers are used in traditional systems
of measurement, and tend to be used in engineering designs, due to
their ease of use in calculations involving vulgar fractions. If
Q(x) denotes the number of highly composite numbers which are less
than or equal to x, then there exist two constants a and b, both
bigger than 1, so that (ln x)a Q(x) (ln x)b. with the first part of
the inequality proved by Paul Erds in 1944 and the second part by
Jean-Louis Nicolas in 1988.
98
ExamplesThe highly composite number : 10,080.10,080 = (2 2 2 2
2) (3 3) 5 7 By (2) above, 10,080 has exactly seventy-two divisors.
1 10,080 7 1,440 15 672 28 360 42 240 70 144 2 5,040 8 1,260 16 630
30 336 45 224 72 140 3 3,360 9 1,120 18 560 32 315 48 210 80 126 4
2,520 10 1,008 20 504 35 288 56 180 84 120 5 2,016 12 840 21 480 36
280 60 168 90 112 6 1,680 14 720 24 420 40 252 63 160 96 105
Highly composite number
99
Note: The bolded numbers are themselves highly composite
numbers. Only the twentieth highly composite number 7560 (=32520)
is absent. 10080 is a so-called 7-smooth number, (sequence A002473
[4] in OEIS).
The 15,000th highly composite number can be found on Achim
Flammenkamp's website. It is the product of 230 primes: , where is
the sequence of successive prime numbers, and all omitted terms
(a22 to a228) are factors with exponent equal to one (i.e. the
number is ). [5]
Prime factor subsetsFor any highly composite number, if one
takes any subset of prime factors for that number and their
exponents, the resulting number will have more divisors than any
smaller number that uses the same prime factors. For example for
the highly composite number 720 which is 24325 we can be sure that
144 which is 2432 has more divisors than any smaller number that
has only the prime factors 2 and 3 80 which is 245 has more
divisors than any smaller number that has only the prime factors 2
and 5 45 which is 325 has more divisors than any smaller number
that has only the prime factors 3 and 5 If this were untrue for any
particular highly composite number and subset of prime factors, we
could exchange that subset of primefactors and exponents for the
smaller number using the same primefactors and get a smaller number
with at least as many divisors. This property is useful for finding
highly composite numbers.
References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa002182
[2] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa005179 [3]
Srinivasan, A. K. (1948), "Practical numbers" (http:/ / www. ias.
ac. in/ jarch/ currsci/ 17/ 179. pdf), Current Science 17: 179180,
MR0027799, . [4] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa002473
[5] Flammenkamp, Achim, Highly Composite Numbers (http:/ /
wwwhomes. uni-bielefeld. de/ achim/ highly. html), .
External links Weisstein, Eric W., " Highly Composite Number
(http://mathworld.wolfram.com/HighlyCompositeNumber. html)" from
MathWorld. Earth360: Versatile Numbers: Self-Organization,
Emergence, and Economics (http://earth360.com/ math-versatile.html)
Algorithm for computing Highly Composite Numbers
(http://web.archive.org/web/19980707133810/www.
math.princeton.edu/~kkedlaya/math/hcn-algorithm.tex) First 10000
Highly Composite Numbers
(http://web.archive.org/web/19980707133953/www.math.
princeton.edu/~kkedlaya/math/hcn10000.txt.gz) Achim Flammenkamp,
First 779674 HCN with sigma,tau,factors
(http://wwwhomes.uni-bielefeld.de/achim/ highly.html)
Home prime
100
Home primeIn number theory, the home prime HP(n) of an integer n
greater than 1 is the prime obtained by repeatedly factoring the
increasing concatenation of prime factors including repetitions.
The mth intermediate stage in the process of determining HP(n) is
designated HPn(m). For instance, HP(10)=773, as 10 factors as 25
yielding HP10(1)=25, 25 factors as 55 yielding HP10(2)=HP25(1)=55,
55=511 implies HP10(3)=HP25(2)=HP55(1)=511, and 511=773 gives
HP10(4)=HP25(3)=HP55(2)=HP511(1)=773, a prime number. Some sources
use the alternative notation HPn for the homeprime, leaving out
parentheses. Investigations into home primes make up a minor side
issue in number theory. Its questions have served as test fields
for the implementation of efficient algorithms for factoring
composite numbers, but the subject is really one in recreational
mathematics. The outstanding computational problem at present
(January, 2011) is whether HP(49)=HP(77) can be calculated in
practice. As each iteration is greater than the previous up until a
prime is reached, factorizations generally grow more difficult so
long as an end is not reached. The pursuit of HP(49) is now a
process of factoring a 210-digit composite factor of HP49(106)
after a break was achieved on 8 February 2010 with the calculation
of HP49(100) after work lasting the larger part of a decade and an
extensive use of computational resources and the successful
factorization of a 178-digit composite in HP49(104) into 88- and
90-digit primes on 11 January 2011. Details of the history of this
search, as well as the sequences leading to home primes for all
other numbers through 100, are maintained at Patrick De Geest's
worldofnumbers website. A wiki primarily associated with the Great
Internet Mersenne Prime Search maintains the complete known data
through 1000 in base 10 and also has lists for the bases 2
through9. Aside from the computational problems that have had so
much time devoted to them, it appears absolute proof of existence
of a home prime for any specific number might entail its effective
computation. In purely heuristic terms, the existence has
probability 1 for all numbers, but such heuristics make assumptions
about numbers drawn from a wide variety of processes that, though
they likely are correct, fall short of the standard of proof
usually required of mathematical claims.
Early history and additional terminologyWhile it's unlikely that
the idea was not conceived of numerous times in the past, the first
reference in print appears to be an article written in 1990 in a
small and now-defunct publication called Recreational and
Educational Computation. The same person who authored that article,
Jeffrey Heleen, revisited the subject in the 19967 volume of the
Journal of Recreational Mathematics in an article entitled Family
Numbers: Constructing Primes By Prime Factor Splicing, which
included all of the results HP(n) for n through 100 other than the
ones still unresolved. It also included a now-obsolete list of
3-digit unresolved numbers (The 58 listed have been cut precisely
in half as of January, 2011). It appears that this article is
largely responsible for provoking attempts by others to resolve the
case involving 49 and77. The article uses the terms daughter and
parent to describe composites and the primes that they lead to,
with numbers leading to the same home prime called siblings (even
if one is an iterate of another), and calls the number of
iterations required to reach a parent, the persistence of a number
under the map to obtain a home prime, the number of lives. The
brief article does little other than state the origins of the
subject, define terms, give a couple of examples, mention machinery
and methods used at the time, and then provide tables. It appears
that Mr. De Geest is responsible for the notation now in use. The
OEIS also uses homeliness as the term for the number of numbers,
including the prime itself, that have a certain prime as its home
prime.
Home prime
101
References1. http://oeis.org/A037274 2.
http://www.worldofnumbers.com/topic1.htm 3.
http://mathworld.wolfram.com/HomePrime.html 4.
http://www.mersennewiki.org/index.php/Home_Primes_Search 5. J.
Heleen, Family Numbers: Constructing Primes By Prime Factor
Splicing, J. Rec. Math., 28, pp.1169, 1996-7 6. J. Heleen, Family
Numbers: Mathemagical Black Holes, Recreational and Educational
Computing, 5:5, p.6, 1990
Hyperperfect numberIn mathematics, a k-hyperperfect number is a
natural number n for which the equality n = 1 + k((n) n 1) holds,
where (n) is the divisor function (i.e., the sum of all positive
divisors of n). A hyperperfect number is a k-hyperperfect number
for some integer k. Hyperperfect numbers generalize perfect
numbers, which are 1-hyperperfect. The first few numbers in the
sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496,
... (sequence A034897 [1] in OEIS), with the corresponding values
of k being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 [2] in
OEIS). The first few k-hyperperfect numbers that are not perfect
are 21, 301, 325, 697, 1333, ... (sequence A007592 [3] in
OEIS).
List of hyperperfect numbersThe following table lists the first
few k-hyperperfect numbers for some values of k, together with the
sequence number in the On-Line Encyclopedia of Integer Sequences
(OEIS) of the sequence of k-hyperperfect numbers:k 1 2 3 4 6 10 11
12 18 19 30 31 35 40 48 59 60 A028500 A028501 [7] [8] A028499 [6]
OEIS A000396 A007593 [4] [5] Some known k-hyperperfect numbers 6,
28, 496, 8128, 33550336, ... 21, 2133, 19521, 176661, 129127041,
... 325, ... 1950625, 1220640625, ... 301, 16513, 60110701,
1977225901, ... 159841, ... 10693, ... 697, 2041, 1570153,
62722153, 10604156641, 13544168521, ... 1333, 1909, 2469601,
893748277, ... 51301, ... 3901, 28600321, ... 214273, ... 306181,
... 115788961, ... 26977, 9560844577, ... 1433701, ... 24601,
...
Hyperperfect number
10266 75 78 91 100 108 126 132 136 138 140 168 174 180 190 192
198 206 222 228 252 276 282 296 342 348 350 360 366 372 396 402 408
414 430 438 480 522 546 296341, ... 2924101, ... 486877, ...
5199013, ... 10509080401, ... 275833, ... 12161963773, ... 96361,
130153, 495529, ... 156276648817, ... 46727970517, 51886178401, ...
1118457481, ... 250321, ... 7744461466717, ... 12211188308281, ...
1167773821, ... 163201, 137008036993, ... 1564317613, ...
626946794653, 54114833564509, ... 348231627849277, ... 391854937,
102744892633, 3710434289467, ... 389593, 1218260233, ...
72315968283289, ... 8898807853477, ... 444574821937, ... 542413,
26199602893, ... 66239465233897, ... 140460782701, ... 23911458481,
... 808861, ... 2469439417, ... 8432772615433, ... 8942902453,
813535908179653, ... 1238906223697, ... 8062678298557, ...
124528653669661, ... 6287557453, ... 1324790832961, ...
723378252872773, 106049331638192773, ... 211125067071829, ...
Hyperperfect number
103570 660 672 684 774 810 814 816 820 968 972 978 1050 1410
2772 3918 9222 9828 14280 23730 31752 55848 67782 92568 100932
A034916 A028502 [9] 1345711391461, 5810517340434661, ...
13786783637881, ... 142718568339485377, ... 154643791177, ...
8695993590900027, ... 5646270598021, ... 31571188513, ...
31571188513, ... 1119337766869561, ... 52335185632753, ...
289085338292617, ... 60246544949557, ... 64169172901, ...
80293806421, ... 95295817, 124035913, ... 61442077, 217033693,
12059549149, 60174845917, ... 404458477, 3426618541, 8983131757,
13027827181, ... 432373033, 2797540201, 3777981481, 13197765673,
... 848374801, 2324355601, 4390957201, 16498569361, ... 2288948341,
3102982261, 6861054901, 30897836341, ... [10] 4660241041,
7220722321, 12994506001, 52929885457, 60771359377, ... 15166641361,
44783952721, 67623550801, ... 18407557741, 18444431149,
34939858669, ... 50611924273, 64781493169, 84213367729, ...
50969246953, 53192980777, 82145123113, ...
It can be shown that if k > 1 is an odd integer and p = (3k +
1) / 2 and q = 3k + 4 are prime numbers, then pq is k-hyperperfect;
Judson S. McCranie has conjectured in 2000 that all k-hyperperfect
numbers for odd k > 1 are of this form, but the hypothesis has
not been proven so far. Furthermore, it can be proven that if p q
are odd primes and k is an integer such that k(p + q) = pq - 1,
then pq is k-hyperperfect. It is also possible to show that if k
> 0 and p = k + 1 is prime, then for all i > 1 such that q =
pi p + 1 is prime, n = pi 1 q is k-hyperperfect. The following
table lists known values of k and corresponding values of i for
which n is k-hyperperfect:
Hyperperfect number
104
k 16 22 28 36 42 46 52 58 72 88 96
OEIS A034922
Values of i
[11] 11, 21, 127, 149, 469, ... 17, 61, 445, ... 33, 89, 101,
... 67, 95, 341, ...
A034923 A034924
[12] 4, 6, 42, 64, 65, ... [13] 5, 11, 13, 53, 115, ... 21, 173,
... 11, 117, ... 21, 49, ...
A034925
[14] 9, 41, 51, 109, 483, ... 6, 11, 34, ...
100 A034926 [15] 3, 7, 9, 19, 29, 99, 145, ...
HyperdeficiencyThe newly-introduced mathematical concept of
hyperdeficiency is related to the hyperperfect numbers. Definition
(Minoli 2010): For any integer n and for integer k, - 0.
Further readingArticles Daniel Minoli, Robert Bear (Fall 1975),
"Hyperperfect numbers", Pi Mu Epsilon Journal 6 (3): 153157. Daniel
Minoli (Dec 1978), "Sufficient forms for generalized perfect
numbers", Annales de la Facult des Sciences UNAZA 4 (2): 277302.
Daniel Minoli (Feb. 1981), "Structural issues for hyperperfect
numbers", Fibonacci Quarterly 19 (1): 614. Daniel Minoli (April
1980), "Issues in non-linear hyperperfect numbers", Mathematics of
Computation 34 (150): 639645. Daniel Minoli (October 1980), "New
results for hyperperfect numbers", Abstracts of the American
Mathematical Society 1 (6): 561. Daniel Minoli, W. Nakamine (1980),
"Mersenne numbers rooted on 3 for number theoretic transforms",
International Conference on Acoustics, Speech, and Signal
Processing. Judson S. McCranie (2000), "A study of hyperperfect
numbers" [16], Journal of Integer Sequences 3.
Hyperperfect number
105
Books Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY,
2002, ISBN 0-07-140615-8 (p.114-134)
External links MathWorld: Hyperperfect number [17] has a long
list of hyperperfect numbers under Data [18]
References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034897
[2] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034898 [3] http:/ /
en. wikipedia. org/ wiki/ Oeis%3Aa007592 [4] http:/ / en.
wikipedia. org/ wiki/ Oeis%3Aa000396 [5] http:/ / en. wikipedia.
org/ wiki/ Oeis%3Aa007593 [6] http:/ / en. wikipedia. org/ wiki/
Oeis%3Aa028499 [7] http:/ / en. wikipedia. org/ wiki/
Oeis%3Aa028500 [8] http:/ / en. wikipedia. org/ wiki/
Oeis%3Aa028501 [9] http:/ / en. wikipedia. org/ wiki/
Oeis%3Aa028502 [10] http:/ / en. wikipedia. org/ wiki/
Oeis%3Aa034916 [11] [12] [13] [14] [15] [16] [17] [18] http:/ / en.
wikipedia. org/ wiki/ Oeis%3Aa034922 http:/ / en. wikipedia. org/
wiki/ Oeis%3Aa034923 http:/ / en. wikipedia. org/ wiki/
Oeis%3Aa034924 http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034925
http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034926 http:/ / www.
math. uwaterloo. ca/ JIS/ VOL3/ mccranie. html http:/ / mathworld.
wolfram. com/ HyperperfectNumber. html http:/ / j. mccranie. home.
comcast. net
Juggler sequence
106
Juggler sequenceIn recreational mathematics a juggler sequence
is an integer sequence that starts with a positive integer a0, with
each subsequent term in the sequence defined by the recurrence
relation:
BackgroundJuggler sequences were publicised by American
mathematician and author Clifford A. Pickover.[1] The name is
derived from the rising and falling nature of the sequences, like
balls in the hands of a juggler.[2] For example, the juggler
sequence starting with a0 = 3 is
If a juggler sequence reaches 1, then all subsequent terms are
equal to 1. It is conjectured that all juggler sequences eventually
reach 1. This conjecture has been verifed for initial terms up to
106,[3] but has not been proved. Juggler sequences therefore
present a problem that is similar to the Collatz conjecture, about
which Paul Erds stated that "mathematics is not yet ready for such
problems". For a given initial term n we define l(n) to be the
number of steps which the juggler sequence starting at n takes to
first reach 1, and h(n) to be the maximum value in the juggler
sequence starting at n. For small values of n we have:n 2 3 4 5 6 7
8 9 Juggler sequence 2, 1 3, 5, 11, 36, 6, 2, 1 4, 2, 1 5, 11, 36,
6, 2, 1 6, 2, 1 7, 18, 4, 2, 1 8, 2, 1 9, 27, 140, 11, 36, 6, 2, 1
l(n) (sequence A007320 1 6 2 5 2 4 2 7 7 [4] in OEIS) h(n)
(sequence A094716 2 36 4 36 6 18 8 140 36 [5] in OEIS)
10 10, 3, 5, 11, 36, 6, 2, 1
Juggler sequences can reach very large values before descending
to 1. For example, the juggler sequence starting at a0 = 37 reaches
a maximum value of 24906114455136. Harry J. Smith has determined
that the juggler sequence starting at a0 = 48443 reaches a maximum
value at a60 with 972,463 digits, before reaching 1 at a157.[6]
Juggler sequence
107
References[1] [2] [3] [4] [5] [6] Pickover, Clifford A. (1992).
Computers and the Imagination. St. Martin's Press. pp.Chapter 40.
ISBN978-0312083434. Pickover, Clifford A. (2002). The Mathematics
of Oz. Cambridge University Press. pp.Chapter 45.
ISBN978-0521016780. *Weisstein, Eric W., " Juggler Sequence (http:/
/ mathworld. wolfram. com/ JugglerSequence. html)" from MathWorld.
http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa007320 http:/ / en.
wikipedia. org/ wiki/ Oeis%3Aa094716 Letter from Harry J. Smith to
Cliiford A. Pickover, 27th June 1992 (http:/ / web. archive. org/
web/ 20091027155431/ http:/ / geocities. com/ hjsmithh/ Juggler/
Juggle3L. html)
External links Juggler sequence calculator
(http://members.chello.nl/k.ijntema/juggler.html) at Collatz
Conjecture Calculation Center Juggler Number pages
(http://web.archive.org/web/20091027103635/http://geocities.com/hjsmithh/
Juggler/index.html) by Harry J. Smith
Kolakoski sequenceIn mathematics, the Kolakoski sequence is an
infinite list that begins with
1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,... The
rules for generating this sequence are as follows: 1) The only
numbers allowed are 1 and 2 2) Each number tells us how many
numbers to append to the sequence 2a) 1 tells us to append one more
number 2b) 2 tells us to append two more numbers 3) We can have no
more than two of the same number in sequence 3a) This is because if
we write 222 that means that somewhere in the sequence it told us
to append 3 twos, but the only numbers allowed are 1 and 2 4) Every
time we "read" a new number, we alternate between writing 1 and 2
5) A0=1 This is an example of a self-generating sequence, with the
first term as 1. Each individual number describes how many numbers
to write next. This is called the run of the numbers. To start the
sequence we have to write one 2 after the 1 because the 1 tells us
to write one number, and because our initial condition "wrote" the
one we must alternate by writing the 2. This gives us as our
sequence so far (1 2). Next, The 2 tells us that the length, or the
run, of the next set of numbers must be 2, but we cannot have any
more than two ones or two twos in sequence because that would mean
that the sequence told us to write three 2's, which would mean a 3
would have to be in the sequence, which is not allowed. Therefore,
we must write 2 1. This gives us our sequence so far (1 2 2 1). The
next number in the list is a 2, so we must write two numbers. The
next two numbers that we should write is a 1; however, we cannot
write three 1's in a row, so we must append 1 2 to the sequence.
This gives us our sequence so far (1 2 2 1 1 2). The fourth number
in the list is a 1. As we wrote a 2 last we write a 1 now. This
gives us our sequence so far (1 2 2 1 1 2 1). The fifth number in
the sequence is a 1. As we wrote a 1 before, we must write a 2 now.
This gives us our sequence so far (1 2 2 1 1 2 1 2). This continues
on. Another explanation for the generation of the Kolakoski
sequence is indicated here: (1) write 1; read it as the number of
1's to write before switching to 2;
Kolakoski sequence (2) write 2; read it as the number of 2's to
write before switching back to 1; (3) so far... 1,2,2; read the new
2 as the number of 1's to write; (4) so far... 1,2,2,1,1; read the
new 1,1 as the number of 2's and then 1's to write; (5) so far...
1,2,2,1,1,2,1; continue generating forever. To truly understand how
the sequence is generated, follow either of the two examples above
and write out the first few terms. It seems plausible that the
density of 1's is 1/2, but this conjecture remains unproved. This
and related simply stated unsolved problems are presented at
Integer Sequences and Arrays. [1] Attempts to solve them are
referenced at MathWorld [2] and sequence A000002 [3] at On-Line
Encyclopedia of Integer Sequences. In the unpublished technical
report Notes on the Kolakoski Sequence [4], Chvtal proved that the
upper density of 1's is less than 0.50084. Kolakoski's
self-generating sequence has attracted the interest of computer
scientists as well as mathematicians. For example, in A New Kind of
Science, p. 895, Stephen Wolfram describes the Kolakoski sequence
in connection with the history of cyclic tag systems.
108
Kolakoski fanThe Kolakoski fan is the following array: 1 2 22
1122 121122 211212211 (and so on, where the initial entries are
given by the Kolakoski sequence, and the block lengths in each row
are given by the entries in the previous row) Associated with this
array, indexed as A143477 [5] (in the On-Line Encyclopedia of
Integer Sequences) are many other such arrays, such as A111090 [6],
for which it is conjectured that the growth rate of row-length is
asymptotic to c(3/2)n for some constant c.
References J.-P. Allouche and J. Shallit, Automatic Sequences,
Cambridge Univ. Press, 2003, p. 337. Vaek Chvtal, "Notes on the
Kolakoski Sequence", DIMACS Technical Report 93-84, December 1993.
F. M. Dekking, "What Is the Long Range Order in the Kolakoski
Sequence?" In Proceedings of the NATO Advanced Study Institute held
in Waterloo, ON, August 21-September 1, 1995 (dd. R. V. Moody).
Dordrecht, Netherlands: Kluwer, pp. 115-125, 1997. M. S. Keane,
"Ergodic Theory and Subshifts of Finite Type." In Ergodic Theory,
Symbolic Dynamics and Hyperbolic Spaces (ed. T. Bedford and M.
Keane). Oxford, England: Oxford University Press, pp. 35-70, 1991.
William Kolakoski, proposal 5304, American Mathematical Monthly 72
(1965), 674; for a partial solution, see "Self Generating Runs," by
Necdet oluk, Amer. Math. Mon. 73 (1966), 681-2. J. C. Lagarias,
"Number Theory and Dynamical Systems." In The Unreasonable
Effectiveness of Number Theory (ed. S. A. Burr). Providence, RI:
Amer. Math. Soc., pp. 35-72, 1992. G. Paun and A. Salomaa,
"Self-Reading Sequences." Amer. Math. Monthly 103, 166-168,
1996.
Kolakoski sequence Bertran Steinsky, "A Recursive Formula for
the Kolakoski Sequence A000002," Journal of Integer Sequences 9
(2006) Article 06.3.7. J.M. Fedou and G. Fici, "Some remarks on
differentiable sequences and recursivity", "Journal of Integer
Sequences" 13(3) (2010) Article 10.3.2.
109
External links Kolakoski Sequence [2] Online Encyclopedia of
Integer Sequences [7] (search "Kolakoski") Kolakoski Constant to
25000 Digits. [8]
References[1] [2] [3] [4] [5] [6] [7] http:/ / faculty.
evansville. edu/ ck6/ integer/ index. html http:/ / mathworld.
wolfram. com/ KolakoskiSequence. html http:/ / en. wikipedia. org/
wiki/ Oeis%3Aa000002 http:/ / dimacs. rutgers. edu/
TechnicalReports/ abstracts/ 1993/ 93-84. html http:/ / en.
wikipedia. org/ wiki/ Oeis%3Aa143477 http:/ / en. wikipedia. org/
wiki/ Oeis%3Aa111090 http:/ / www. research. att. com/ ~njas/
sequences/
[8] http:/ / pi. lacim. uqam. ca/ piDATA/ Kolakoski. txt
Lucky numberIn number theory, a lucky number is a natural number
in a set which is generated by a "sieve" similar to the Sieve of
Eratosthenes that generates the primes. Begin with a list of
integers starting with 1: 1, 2, 3, 4, 5, 6, 7, 8,
9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, Every second
number (all even numbers) is eliminated, leaving only the odd
integers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25,
The second term in this sequence is 3. Every third number which
remains in the list is eliminated: 1, 3, 7, 9, 13, 15, 19, 21,
25,
The third surviving number is now 7, so every seventh number
that remains is eliminated: 1, 3, 7, 9, 13, 15, 21, 25,
As this procedure is repeated indefinitely, the survivors are
the lucky numbers: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49,
51, 63, 67, 69, 73, 75, 79, 87, 93, 99, ... (sequence A000959 [1]
in OEIS).
Lucky number
110
The term was introduced in 1955 in a paper by Gardiner, Lazarus,
Metropolis and Ulam. They suggest also calling its defining sieve
the sieve of Josephus Flavius[2] because of its similarity with the
counting-out game in the Josephus problem. Lucky numbers share some
properties with primes, such as asymptotic behaviour according to
the prime number theorem; also Goldbach's conjecture has been
extended to them. There are infinitely many lucky numbers. Because
of these apparent connections with the prime numbers, some
mathematicians have suggested that these properties may be found in
a larger class of sets of numbers generated by sieves of a certain
unknown form, although there is little theoretical basis for this
conjecture. Twin lucky numbers and twin primes also appear to occur
with similar frequency. A lucky prime is a lucky number that is
prime. It is not known whether there are infinitely many lucky
primes. The first few are 3, 7, 13, 31, 37, 43, 67, 73, 79, 127,
151, 163, 193 (sequence A031157 [3] in OEIS).An animation
demonstrating the lucky number sieve. The numbers in red are lucky
numbers.
References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000959
[2] V. Gardiner, R. Lazarus, N. Metropolis and S. Ulam, "On certain
sequences of integers defined by sieves", Mathematics Magazine 29:3
(1955), pp. 117122. [3] http:/ / en. wikipedia. org/ wiki/
Oeis%3Aa031157
External links Peterson, Ivars. MathTrek: Martin Gardner's Lucky
Number (http://www.sciencenews.org/sn_arc97/9_6_97/ mathland.htm)
Weisstein, Eric W., " Lucky Number
(http://mathworld.wolfram.com/LuckyNumber.html)" from MathWorld.
Lucky Numbers (http://demonstrations.wolfram.com/LuckyNumbers/) by
Enrique Zeleny, The Wolfram Demonstrations Project.
Lucas number
111
Lucas numberThe Lucas numbers are an integer sequence named
after the mathematician Franois douard Anatole Lucas (18421891),
who studied both that sequence and the closely related Fibonacci
numbers. Lucas numbers and Fibonacci numbers form complementary
instances of Lucas sequences.
DefinitionLike the Fibonacci numbers, each Lucas number is
defined to be the sum of its two immediate previous terms, i.e. it
is a Fibonacci integer sequence. Consequently, the ratio between
two consecutive Lucas numbers converges to the golden ratio.
However, the first two Lucas numbers are L0 = 2 and L1 = 1 instead
of 0 and 1, and the properties of Lucas numbers are therefore
somewhat different from those of Fibonacci numbers. A Lucas number
may thus be defined as follows:
The sequence of Lucas numbers begins: 2, 1, 3, 4, 7, 11, 18, 29,
47, 76, 123, ... (sequence A000032 [1] in OEIS)
Extension to negative integersUsing Ln-2 = Ln - Ln-1, one can
extend the Lucas numbers to negative integers to obtain a doubly
infinite sequence : ..., -11, 7, -4, 3, -1, 2, 1, 3, 4, 7, 11, ...
(terms for are shown). The formula for terms with negative indices
in this sequence is
Relationship to Fibonacci numbersThe Lucas numbers are related
to the Fibonacci numbers by the identities Their closed formula is
given as: , and thus as approaches +, the ratio approaches
where
is the Golden ratio. Alternatively,
is the closest integer to
.
Lucas number
112
Congruence relationLn is congruent to 1 mod n if n is prime, but
some composite values of n also have this property.
Lucas primesA Lucas prime is a Lucas number that is prime. The
first few Lucas primes are 2, 3, 7, 11, 29, 47, 199, 521, 2207,
3571, 9349, ... (sequence A005479 [2] in OEIS) If Ln is prime then
n is either 0, prime, or a power of 2.[3] L values of . is prime
for = 1, 2, 3, and 4 and no other known
Lucas polynomialsThe Lucas polynomials Ln(x) are a polynomial
sequence derived from the Lucas numbers in the same way as
Fibonacci polynomials are derived from the Fibonacci numbers. Lucas
polynomials are defined by the following recurrence relation:
Lucas polynomials can be expressed in terms of Lucas sequences
as
The first few Lucas polynomials are:
The Lucas numbers are recovered by evaluating the polynomials at
x=1. The degree of Ln(x) is n. The ordinary generating function for
the sequence is
References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000032
[2] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa005479 [3] Chris
Caldwell, " The Prime Glossary: Lucas prime (http:/ / primes. utm.
edu/ glossary/ page. php?sort=LucasPrime)" from The Prime
Pages.
External links Weisstein, Eric W., " Lucas Number
(http://mathworld.wolfram.com/LucasNumber.html)" from MathWorld.
Weisstein, Eric W., " Lucas Polynomial
(http://mathworld.wolfram.com/LucasPolynomial.html)" from
MathWorld. Dr Ron Knott
(http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucasNbs.html)
Lucas numbers and the Golden Section
(http://milan.milanovic.org/math/english/lucas/lucas.html)
Lucas number A Lucas Number Calculator can be found here.
(http://www.plenilune.pwp.blueyonder.co.uk/
fibonacci-calculator.asp) A Tutorial on Generalized Lucas Numbers
(http://nakedprogrammer.com/LucasNumbers.aspx)
113
Padovan sequenceThe Padovan sequence is the sequence of integers
P(n) defined by the initial values and the recurrence relation
The first few values of P(n) are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9,
12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... (sequence
A000931 [1] in OEIS) The Padovan sequence is named after Richard
Padovan who attributed its discovery to Dutch architect Hans van
der Laan in his 1994 essay Dom. Hans van der Laan : Modern
Primitive. The sequence was described by Ian Stewart in his
Scientific American column Mathematical Recreations in June 1996.
The above definition is the one given by Ian Stewart and by
MathWorld. Other sources may start the sequence at a different
place, in which case some of the identities in this article must be
adjusted with appropriate offsets.
Spiral of equilateral triangles with side lengths which follow
the Padovan sequence.
Recurrence relationsIn the spiral, each triangle shares a side
with two others giving a visual proof that the Padovan sequence
also satisfies the recurrence relation
Starting from this, the defining recurrence and other
recurrences as they are discovered, one can create an infinite
number of further recurrences by repeatedly replacing by The Perrin
sequence satisfies the same recurrence relations as the Padovan
sequence, although it has different initial values. This is a
property of recurrence relations. The Perrin sequence can be
obtained from the Padovan sequence by the following formula:
Padovan sequence
114
Extension to negative parametersAs with any sequence defined by
a recurrence relation, Padovan numers P(m) for m n p(k, n) = 1 if k
= n p(k, n) = p(k+1, n) + p(k, n k) otherwise. This function tends
to exhibit deceptive behavior. p(1, 4) = 5 p(2, 8) = 7 p(3, 12) = 9
p(4, 16) = 11 p(5, 20) = 13 p(6, 24) = 16 Our original function
p(n) is just p(1, n). The values of this function:k 1 n 1 2 3 4 5 6
7 8 9 10 1 2 3 5 7 11 15 22 30 2 0 1 1 2 2 4 4 7 8 3 0 0 1 1 1 2 2
3 4 5 4 0 0 0 1 1 1 1 2 2 3 5 0 0 0 0 1 1 1 1 1 2 6 0 0 0 0 0 1 1 1
1 1 7 0 0 0 0 0 0 1 1 1 1 8 0 0 0 0 0 0 0 1 1 1 9 0 0 0 0 0 0 0 0 1
1 10 0 0 0 0 0 0 0 0 0 1
42 12
Partition number
120
Generating functionA generating function for p(n) is given by
the reciprocal of Euler's function:
Expanding each term on the right-hand side as a geometric
series, we can rewrite it as (1 + x + x2 + x3 + ...)(1 + x2 + x4 +
x6 + ...)(1 + x3 + x6 + x9 + ...) .... The xn term in this product
counts the number of ways to write n = a1 + 2a2 + 3a3 + ... = (1 +
1 + ... + 1) + (2 + 2 + ... + 2) + (3 + 3 + ... + 3) + ..., where
each number i appears ai times. This is precisely the definition of
a partition of n, so our product is the desired generating
function. More generally, the generating function for the
partitions of n into numbers from a set A can be found by taking
only those terms in the product where k is an element of A. This
result is due to Euler. The formulation of Euler's generating
function is a special case of a q-Pochhammer symbol and is similar
to the product formulation of many modular forms, and specifically
the Dedekind eta function. It can also be used in conjunction with
the pentagonal number theorem to derive a recurrence for the
partition function stating that: p(k) = p(k 1) + p(k 2) p(k 5) p(k
7) + p(k 12) + p(k 15) p(k 22) ... where p(0) is taken to equal 1,
p(k) is zero for negative k, and the sum is taken over all
generalized pentagonal numbers of the form n(3n 1), for n running
over positive and negative integers: successively taking n = 1, 1,
2, 2, 3, 3, 4, 4 ..., generates the values 1, 2, 5, 7, 12, 15, 22,
26, 35, 40, 51, .... The signs in the summation continue to
alternate+,+,,,+,+,...
Table of valuesSome values of the partition function are as
follows (sequence A000041 [2] in OEIS): p(1) = 1 p(2) = 2 p(3) = 3
p(4) = 5 p(5) = 7 p(6) = 11 p(7) = 15 p(8) = 22 p(9) = 30 p(10) =
42 p(100) = 190,569,292 p(200) = 3,972,999,029,388 p(1000) =
24,061,467,864,032,622,473,692,149,727,991 2.41031.
As of February 2010, the largest known prime number of this kind
is p(29099391), with 6002 decimal digits, found by Predrag
Minovic.[3]
Partition number
121
Asymptotic behaviourAn asymptotic expression for p(n) is given
by
This asymptotic formula was first obtained by G. H. Hardy and
Ramanujan in 1918 and independently by J. V. Uspensky in 1920.
Considering p(1000), the asymptotic formula gives about 2.44021031,
reasonably close to the exact answer given above. In 1937, Hans
Rademacher was able to improve on Hardy and Ramanujan's results by
providing a convergent series expression for p(n). It is
where
Here, the notation (m,n)=1 implies that the sum should occur
only over the values of m that are relatively prime to n. The
function s(m,k) is a Dedekind sum. The proof of Rademacher's
formula is interesting in that it involves Ford circles, Farey
sequences, modular symmetry and the Dedekind eta function in a
central way.
CongruencesSrinivasa Ramanujan is credited with discovering that
"congruences" in the number of partitions exist for integers ending
in 4 and 9.
For instance, the number of partitions for the integer 4 is 5.
For the integer 9, the number of partitions is 30; for 14 there are
135 partitions. He also discovered congruences related to 7 and
11:
Since 5, 7, and 11 are consecutive primes, one might think that
there would be such a congruence for the next prime 13, for some a.
This is, however, false. It can also be shown that there is no
congruence of the form for any prime b other than 5, 7, or 11. In
the 1960s, A. O. L. Atkin of the University of Illinois at Chicago
discovered additional congruences for small prime moduli. For
example:
In 2000, Ken Ono of the University of WisconsinMadison proved
that there are such congruences for every prime modulus. A few
years later Ono, together with Scott Ahlgren of the University of
Illinois, proved that there are partition congruences modulo every
integer coprime to 6.[4]
Partition number
122
Restricted partitionsAmong the 22 partitions for the number 8, 6
contain only odd parts: 7+1 5+3 5+1+1+1 3+3+1+1 3+1+1+1+1+1
1+1+1+1+1+1+1+1
If we count the partitions of 8 with distinct parts, we also
obtain the number 6: 8 7+1 6+2 5+3 5+2+1 4+3+1
It is true for all positive numbers that the number of
partitions with odd parts always equals the number of partitions
with distinct parts. This result was proved by Leonard Euler in
1748.[5] Some similar results about restricted partitions can be
obtained by the aid of a visual tool, a Ferrers graph (also called
Ferrers diagram, since it is not a graph in the graph-theoretical
sense, or sometimes Young diagram, alluding to the Young
tableau).
Ferrers diagramThe partition 6+4+3+1 of the positive number 14
can be represented by the following diagram; these diagrams are
named in honor of Norman Macleod Ferrers:
6+4+3+1
The 14 circles are lined up in 4 columns, each having the size
of a part of the partition. The diagrams for the 5 partitions of
the number 4 are listed below:
4
= 3+1 =
2+2
= 2+1+1 =
1+1+1+1
If we now flip the diagram of the partition 6 + 4 + 3 + 1 along
its main diagonal, we obtain another partition of 14:
Partition number
123
6+4+3+1
=
4+3+3+2+1+1
By turning the rows into columns, we obtain the partition
4+3+3+2+1+1 of the number 14. Such partitions are said to be
conjugate of one another. In the case of the number 4, partitions 4
and 1+1+1+1 are conjugate pairs, and partitions 3+1 and 2+1+1 are
conjugate of each other. Of particular interest is the partition
2+2, which has itself as conjugate. Such a partition is said to be
self-conjugate. Claim: The number of self-conjugate partitions is
the same as the number of partitions with distinct odd parts. Proof
(outline): The crucial observation is that every odd part can be
"folded" in the middle to form a self-conjugate diagram:
One can then obtain a bijection between the set of partitions
with distinct odd parts and the set of self-conjugate partitions,
as illustrated by the following example:
9+7+3 Dist. odd
=
5+5+4+3+2 self-conjugate
Similar techniques can be employed to establish, for example,
the following equalities: The number of partitions of n into no
more than k parts is the same as the number of partitions of n into
parts no larger thank. The number of partitions of n into no more
than k parts is the same as the number of partitions of n+k into
exactly k parts.
Partition number
124
Notes[1] [2] [3] [4] http:/ / www. aimath. org/ news/ partition/
http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000041 http:/ / primes.
utm. edu/ top20/ page. php?id=54 Ono, Ken; Ahlgren, Scott (2001).
"Congruence properties for the partition function" (http:/ / www.
math. wisc. edu/ ~ono/ reprints/ 061. pdf). Proceedings of the
National Academy of Sciences 98 (23): 12,88212,884.
doi:10.1073/pnas.191488598. . [5] Andrews, George E. Number Theory.
W. B. Saunders Company, Philadelphia, 1971. Dover edition, page
149150.
References George E. Andrews, The Theory of Partitions (1976),
Cambridge University Press. ISBN 0-521-63766-X . T