On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009, and is its chairman.
OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited., it contains over 390,000 sequences, and is growing by approximately 30 entries per day.
Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. There is also an advanced search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input.
History
started collecting integer sequences as a graduate student in 1964 to support his work in combinatorics. The database was at first stored on punched cards. He published selections from the database in book form twice:- A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372.
- The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and assigned M-numbers from M0000 to M5487. The Encyclopedia includes the references to the corresponding sequences in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not.
The database continues to grow at a rate of some 10,000 entries a year.
Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the omnibus database.
In 2004, Sloane celebrated the addition of the 100,000th sequence to the database,, which counts the marks on the Ishango bone. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence,, was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list, following a proposal by OEIS Editor-in-Chief Charles Greathouse to choose a special sequence for A200000. A300000 was defined in February 2018, and by end of January 2023 the database contained more than 360,000 sequences.
Non-integers
Besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences.Sequences of fractions are represented by two sequences : the sequence of numerators and the sequence of denominators. For example, the fifth-order Farey sequence,, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5.
Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions ), binary expansions ), or continued fraction expansions ).
Conventions
The OEIS was limited to plain ASCII text until 2011, and it still uses a linear form of conventional mathematical notation. Greek letters are usually represented by their full names, e.g., mu for μ, phi for φ.Every sequence is identified by the letter A followed by six digits, almost always referred to with leading zeros, e.g., A000315 rather than A315.
Individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces.
In comments, formulas, etc.,
a represents the nth term of the sequence.Special meaning of zero
is often used to represent non-existent sequence elements. For example, enumerates the "smallest prime of n2 consecutive primes to form an n × n magic square of least magic constant, or 0 if no such magic square exists." The value of a is 2; a is 1480028129. But there is no such 2 × 2 magic square, so a is 0. This special usage has a solid mathematical basis in certain counting functions; for example, the totient valence function Nφ counts the solutions of φ = m. There are 4 solutions for 4, but no solutions for 14, hence a of A014197 is 0—there are no solutions.Other values are also used, most commonly −1.
Lexicographical ordering
The OEIS maintains the lexicographical order of the sequences, so each sequence has a predecessor and a successor. OEIS normalizes the sequences for lexicographical ordering, ignoring all initial zeros and ones, and also the sign of each element. Sequences of weight distribution codes often omit periodically recurring zeros.For example, consider: the prime numbers, the palindromic primes, the Fibonacci sequence, the lazy caterer's sequence, and the coefficients in the series expansion of. In OEIS lexicographic order, they are:
- Sequence #1: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,...
- Sequence #2: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929,...
- Sequence #3: 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,...
- Sequence #4: 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154,...
- Sequence #5: 3, 8, 3, 24, 24, 48, 3, 8, 72, 120, 24, 168, 144,...
Self-referential sequences
Very early in the history of the OEIS, sequences defined in terms of the numbering of sequences in the OEIS itself were proposed. "I resisted adding these sequences for a long time, partly out of a desire to maintain the dignity of the database, and partly because A22 was only known to 11 terms!", Sloane reminisced.One of the earliest self-referential sequences Sloane accepted into the OEIS was "a = n-th term of sequence An or −1 if An has fewer than n terms". This sequence spurred progress on finding more terms of.
lists the first term given in sequence An, but it needs to be updated from time to time because of changing opinions on offsets. Listing instead term a of sequence An might seem a good alternative if it were not for the fact that some sequences have offsets of 2 and greater.
This line of thought leads to the question "Does sequence An contain the number n?" and the sequences, "Numbers n such that OEIS sequence An contains n", and, "n is in this sequence if and only if n is not in sequence An". Thus, the composite number 2808 is in A053873 because is the sequence of composite numbers, while the non-prime 40 is in A053169 because it is not in, the prime numbers. Each n is a member of exactly one of these two sequences, and in principle it can be determined which sequence each n belongs to, with two exceptions :
- It cannot be determined whether 53873 is a member of A053873 or not. If it is in the sequence then by definition it should be; if it is not in the sequence then it should not be. Nevertheless, either decision would be consistent, and would also resolve the question of whether 53873 is in A053169.
- It can be proved that 53169 both is and is not a member of A053169. If it is in the sequence then by definition it should not be; if it is not in the sequence then it should be. This is a form of Russell's paradox. Hence it is also not possible to answer if 53169 is in A053873.
Abridged example of a comprehensive entry
A046970 Dirichlet inverse of the Jordan function J_2.
1, -3, -8, -3, -24, 24, -48, -3, -8, 72, -120, 24, -168, 144, 192, -3, -288, 24, -360, 72, 384, 360, -528, 24, -24, 504, -8, 144, -840, -576, -960, -3, 960, 864, 1152, 24, -1368, 1080, 1344, 72, -1680, -1152, -1848, 360, 192, 1584, -2208, 24, -48, 72, 2304, 504, -2808, 24, 2880, 144, 2880, 2520, -3480, -576
OFFSET 1,2
COMMENTS B = -B**/)*z/z = -B**/) * Sum_ a/j^.
Apart from signs also Sum_ core^2*mu where core is the squarefree part of x. - Benoit Cloitre, May 31 2002
REFERENCES M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48.
LINKS Reinhard Zumkeller, Table of n, a for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 .
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 403-408.
Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 , 2020.
Wikipedia, Riemann zeta function.
FORMULA Multiplicative with a = 1 - p^2.
a = Sum_ mu*d^2.
abs = Product_. - Jon Perry, Aug 24 2010
From Wolfdieter Lang, Jun 16 2011:
Dirichlet g.f.: zeta/zeta.
a = J_*n^2, with the Jordan function J_k, with J_k:=1. See the Apostol reference, p. 48. exercise 17.
a = -A084920. - R. J. Mathar, Aug 28 2011
G.f.: Sum_ mu*k^2*x^k/. - Ilya Gutkovskiy, Jan 15 2017
EXAMPLE a = -8 because the divisors of 3 are and mu*1^2 + mu*3^2 = -8.
a = -3 because the divisors of 4 are and mu*1^2 + mu*2^2 + mu*4^2 = -3.
E.g., a = * = 8*24 = 192. - Jon Perry, Aug 24 2010
G.f. = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 +...
MAPLE Jinvk := proc local a, f, p ; a := 1 ; for f in ifactors do p := op ; a := a* ; end do: a ; end proc:
A046970 := proc Jinvk ; end proc: # R. J. Mathar, Jul 04 2011
MATHEMATICA muDD := MoebiusMu*d^2; Table
Flatten := If,
a := If, Times @@ ]
PROG A046970=sumdiv \\ Benoit Cloitre
a046970 = product. map . ). a027748_row
-- Reinhard Zumkeller, Jan 19 2012
/* Michael Somos, Jan 11 2014 */
CROSSREFS Cf. A007434, A027641, A027642, A063453, A023900.
Cf. A027748.
Sequence in context: A144457 A220138 A146975 * A322360 A058936 A280369
Adjacent sequences: A046967 A046968 A046969 * A046971 A046972 A046973
KEYWORD sign,easy,mult
AUTHOR Douglas Stoll, dougstollemail.msn.com
EXTENSIONS Corrected and extended by Vladeta Jovovic, Jul 25 2001
Additional comments from Wilfredo Lopez, Jul 01 2005