On Line Encyclopedia of Integer Sequences

History

Neil Sloane started collecting integer sequences as a graduate student in 1965 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:

1. A Handbook of Integer Sequences (1973, ISBN 0-12-648550-X), containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372.
2. The Encyclopedia of Integer Sequences with Simon Plouffe (1995, ISBN 0-12-558630-2), containing 5,488 sequences and assigned M-numbers from M0000 to M5487. The Encyclopedia includes the references to the corresponding sequences (which may differ in their few initial terms) in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 (instead of 1 to 2372.) The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not.

These books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service (August 1994), and soon after as a web site (1996). As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. 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 database. In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, A100000, 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 at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, A200000, 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.

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 rationals are represented by two sequences (named with the keyword 'frac'): the sequence of numerators and the sequence of denominators. For example, the fifth order Farey sequence, 1 5 , 1 4 , 1 3 , 2 5 , 1 2 , 3 5 , 2 3 , 3 4 , 4 5 , is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 (A006842) and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 (A006843). Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions (here 3, 1, 4, 1, 5, 9, 2, 6, ... (A000796)), binary expansions (here 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, ... (A004601)), or continued fraction expansions (here 3, 7, 15, 1, 292, 1, ... (A001203)).

Conventions

The OEIS was limited to plain ASCII text until 2011, yet it still uses a linear form of conventional mathematical notation (such as f(n) for functions, n for running variables, etc.). 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, sometimes referred to without the leading zeros, e.g., A315 rather than A000315. Individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces. In comments, formulas, etc., a(n) represents the nth term of the sequence.

Special meaning of zero

Zero is often used to represent non-existent sequence elements. For example, A104157 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(1) (a 1×1 magic square) is 2; a(3) is 1480028129. But there is no such 2×2 magic square, so a(2) is 0. This special usage has a solid mathematical basis in certain counting functions. For example, the totient valence function N_φ(m) (A014197) counts the solutions of φ(x) = m. There are 4 solutions for 4, but no solutions for 14, hence a(14) of A014197 is 0—there are no solutions. Occasionally −1 is used for this purpose instead, as in A094076.

Lexicographical ordering

The OEIS maintains the lexicographical order of the sequences, so each sequence has a predecessor and a successor (its "context"). OEIS normalizes the sequences for lexicographical ordering, (usually) 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 ζ ( n + 2 ) ζ ( n ) . 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, ... A000040

Sequence #2: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, ... A002385

Sequence #3: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ... A000045

Sequence #4: 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, ... A000124

Sequence #5: 1, −3, −8, −3, −24, 24, −48, −3, −8, 72, −120, 24, −168, 144, ... A046970

whereas unnormalized lexicographic ordering would order these sequences thus: #3, #5, #4, #1, #2.

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 A031135 (later A091967) "a(n) = n-th term of sequence A_n". This sequence spurred progress on finding more terms of A000022. Some sequences are both finite and listed in full (keywords "fini" and "full"); these sequences will never be long enough to contain a term that corresponds to their OEIS sequence number. In this case the corresponding term a(n) of A091967 is undefined (the first case is n = 53) A100544 lists the first term given in sequence A_n, but it needs to be updated from time to time because of changing opinions on offsets. Listing instead term a(1) of sequence A_n might seem a good alternative if it weren't for the fact that some sequences have offsets of 2 and greater. This line of thought leads to the question "Does sequence A_n contain the number n ?" and the sequences A053873, "Numbers n such that OEIS sequence A_n contains n", and A053169, "n is in this sequence if and only if n is not in sequence A_n". Thus, the composite number 2808 is in A053873 because A002808 is the sequence of composite numbers, while the non-prime 40 is in A053169 because it's not in A000040, 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 (related to the two sequences themselves):

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 (again, by definition) 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 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.

An abridged example of a typical OEIS entry

This entry, A046970, was chosen because it contains every field an OEIS entry can have.

A046970 Dirichlet inverse of the Jordan function J_2 (A007434). 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 1,2COMMENTS B(n+2) = -B(n)*((n+2)*(n+1)/(4pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4pi^2))*Sum(j=1, infinity) [ a(j)/j^(n+2) ] ...REFERENCES M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.LINKS M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Wikipedia, Riemann zeta function.FORMULA Multiplicative with a(p^e) = 1-p^2. a(n) = Sum_{d|n} mu(d)*d^2. a(n) = product[p prime divides n, p^2-1] (gives unsigned version) [From Jon Perry (jonperrydc(AT)btinternet.com), Aug 24 2010]EXAMPLE a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8. ...MAPLE Jinvk := proc(n, k) local a, f, p ; a := 1 ; for f in ifactors(n)[2] do p := op(1, f) ; a := a*(1-p^k) ; end do: a ; end proc: A046970 := proc(n) Jinvk(n, 2) ; end proc: # R. J. Mathar, Jul 04 2011 MATHEMATICA muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n, 60}] (Lopez) Flatten[Table[{ x = FactorInteger[n]; p = 1; For[i = 1, i <= Length[x], i++, p = p*(x[[i]][[1]]^2 - 1)]; p}, {n, 1, 50, 1}]] [From Jon Perry (jonperrydc(AT)btinternet.com), Aug 24 2010]PROG (PARI) A046970(n)=sumdiv(n, d, d^2*moebius(d)) (Benoit Cloitre)CROSSREFS Cf. A027641 and A027642. Sequence in context: A035292 A144457 A146975 * A058936 A002017 A118582 Adjacent sequences: A046967 A046968 A046969 * A046971 A046972 A046973 KEYWORD sign,multAUTHOR Douglas Stoll, dougstoll(AT)email.msn.comEXTENSIONS Corrected and extended by Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 25 2001 Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005

Entry fields

See Format of OEIS Pages.

ID number

Every sequence in the OEIS has a serial number, a six-digit positive integer, prefixed by A (and zero-padded on the left prior to November 2004). The letter "A" stands for "absolute". Numbers are either assigned by the editor(s) or by an A number dispenser, which is handy for when contributors wish to send in related sequences at once and be able to create cross-references. An A number from the dispenser expires a month from issue if not used. But as the following table of arbitrarily selected sequences show, the rough correspondence holds.Even for sequences in the book predecessors to the OEIS, the ID numbers are not the same. The 1973 Handbook of Integer Sequences contained about 2400 sequences, which were numbered by lexicographic order (the letter N plus 4 digits, zero-padded where necessary), and the 1995 Encyclopedia of Integer Sequences contained 5487 sequences, also numbered by lexicographic order (the letter M plus 4 digits, zero-padded where necessary). These old M and N numbers, as applicable, are contained in the ID number field in parentheses after the modern A number.

Sequence data

The sequence field lists the numbers themselves, or at least about four lines' worth. The sequence field makes no distinction between sequences that are finite but still too long to display and sequences that are infinite. To help make that determination, you need to look at the keywords field for "fini", "full", or "more". To determine to which n the values given correspond, see the offset field, which gives the n for the first term given.

Name

The name field usually contains the most common name for the sequence, and sometimes also the formula. For example, 1, 8, 27, 64, 125, 216, 343, 512, (A000578) is named "The cubes: a(n) = n^3.".

Comments

The comments field is for information about the sequence that doesn't quite fit in any of the other fields. The comments field often points out interesting relationships between different sequences and less obvious applications for a sequence. For example, Lekraj Beedassy in a comment to A000578 notes that the cube numbers also count the "total number of triangles resulting from criss-crossing cevians within a triangle so that two of its sides are each n-partitioned," while Neil Sloane points out the unexpected relationship between centered hexagonal numbers (A003215) and second Bessel polynomials (A001498) in a comment to A003215.

References

References to printed documents (books, papers, ...).

Links

Links, i.e. URLs, to online resources. These may be:

1. references to applicable articles in journals
2. links to the index
3. links to text files which hold the sequence terms (in a two column format) over a wider range of indices than held by the main database lines
4. links to images in the local database directories which often provide combinatorial background related to graph theory
5. others related to computer codes, more extensive tabulations in specific research areas provided by individuals or research groups

Formula

Formulae, recurrences, generating functions, etc. for the sequence.

Example

Some examples of sequence member values.

Maple

Maple code.

Mathematica

Wolfram Language code.

Program

Originally Maple and Mathematica were the preferred programs for calculating sequences in the OEIS, and they both have their own field labels. As of 2016, Mathematica was the most popular choice with 100,000 Mathematica programs followed by 50,000 PARI/GP programs, 35,000 Maple programs, and 45,000 in other languages.As for any other part of the record, if there is no name given, the contribution (here: program) was written by the original submitter of the sequence.

See also

Sequence cross-references originated by the original submitter are usually denoted by "Cf."Except for new sequences, the see also field also includes information on the lexicographic order of the sequence (its "context") and provides links to sequences with close A numbers (A046967, A046968, A046969, A046971, A046972, A046973, in our example). The following table shows the context of our example sequence, A046970:

Keyword

The OEIS has its own standard set of four or five letter keywords that characterize each sequence:

base The results of the calculation depend on a specific positional base. For example, 2, 3, 5, 7, 11, 101, 131, 151, 181 ... A002385 are prime numbers regardless of base, but they are palindromic specifically in base 10. Most of them are not palindromic in binary. Some sequences rate this keyword depending on how they're defined. For example, the Mersenne primes 3, 7, 31, 127, 8191, 131071, ... A000668 does not rate "base" if defined as "primes of the form 2^n - 1". However, defined as "repunit primes in binary," the sequence would rate the keyword "base".

bref "sequence is too short to do any analysis with", for example, A079243, Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.

cofr The sequence represents a continued fraction, for example, continued fraction expansion of e (A003417) or π (A001203).

cons The sequence is a decimal expansion of an important mathematical constant, like e (A001113) or π (A000796).

core A sequence that is of foundational importance to a branch of mathematics, such as the prime numbers (A000040), the Fibonacci sequence (A000045), etc.

dead This keyword used for erroneous sequences that have appeared in papers or books, or for duplicates of existing sequences. For example, A088552 is the same as A000668.

dumb One of the more subjective keywords, for "unimportant sequences," which may or may not directly relate to mathematics. A001355, "Mix digits of pi and e." is one example of the former, and A082390, "Numbers on a computer numpad, read in a spiral." is an example of the latter.

easy The terms of the sequence can be easily calculated. Perhaps the sequence most deserving of this keyword is 1, 2, 3, 4, 5, 6, 7, ... A000027, where each term is 1 more than the previous term. The keyword "easy" is sometimes given to sequences "primes of the form f(m)" where f(m) is an easily calculated function. (Though even if f(m) is easy to calculate for large m, it might be very difficult to determine if f(m) is prime).

eigen A sequence of eigenvalues.

fini The sequence is finite, although it might still contain more terms than can be displayed. For example, the sequence field of A105417 shows only about a quarter of all the terms, but a comment notes that the last term is 3888.

frac A sequence of either numerators or denominators of a sequence of fractions representing rational numbers. Any sequence with this keyword ought to be cross-referenced to its matching sequence of numerators or denominators, though this may be dispensed with for sequences of Egyptian fractions, such as A069257, where the sequence of numerators would be A000012. This keyword should not be used for sequences of continued fractions, cofr should be used instead for that purpose.

full The sequence field displays the complete sequence. If a sequence has the keyword "full", it should also have the keyword "fini". One example of a finite sequence given in full is that of the supersingular primes A002267, of which there are precisely fifteen.

hard The terms of the sequence cannot be easily calculated, even with raw number crunching power. This keyword is most often used for sequences corresponding to unsolved problems, such as "How many n-spheres can touch another n-sphere of the same size?" A001116 lists the first ten known solutions.

hear A sequence with a graph audio deemed to be "particularly interesting and/or beautiful".

less A "less interesting sequence".

look A sequence with a graph visual deemed to be "particularly interesting and/or beautiful".

more More terms of the sequence are wanted. Readers can submit an extension.

mult The sequence corresponds to a multiplicative function. Term a(1) should be 1, and term a(mn) can be calculated by multiplying a(m) by a(n) if m and n are coprime. For example, in A046970, a(12) = a(3)a(4) = -8 × -3.

new For sequences that were added in the last couple of weeks, or had a major extension recently. This keyword is not given a checkbox in the Web form for submitting new sequences, Sloane's program adds it by default where applicable.

nice Perhaps the most subjective keyword of all, for "exceptionally nice sequences."

nonn The sequence consists of nonnegative integers (it may include zeroes). No distinction is made between sequences that consist of nonnegative numbers only because of the chosen offset (e.g., n³, the cubes, which are all positive from n = 0 forwards) and those that by definition are completely nonnegative (e.g., n², the squares).

obsc The sequence is considered obscure and needs a better definition.

probation Sequences that "may be deleted later at the discretion of the editor."

sign Some (or all) of the values of the sequence are negative. The entry includes both a Signed field with the signs and a Sequence field consisting of all the values passed through the absolute value function.

tabf "An irregular (or funny-shaped) array of numbers made into a sequence by reading it row by row." For example, A071031, "Triangle read by rows giving successive states of cellular automaton generated by "rule 62."

tabl A sequence obtained by reading a geometric arrangement of numbers, such as a triangle or square, row by row. The quintessential example is Pascal's triangle read by rows, A007318.

uned The sequence has not been edited but it could be worth including in the OEIS. The sequence may contain computational or typographical errors. Contributors are encouraged to edit these sequences.

unkn "Little is known" about the sequence, not even the formula that produces it. For example, A072036, which was presented to the Internet Oracle to ponder.

walk "Counts walks (or self-avoiding paths)."

word Depends on the words of a specific language. For example, zero, one, two, three, four, five, etc. For example, 4, 3, 3, 5, 4, 4, 3, 5, 5, 4, 3, 6, 6, 8, 8, 7, 7, 9, 8, 8 ... A005589, "Number of letters in the English name of n, excluding spaces and hyphens."

Some keywords are mutually exclusive, namely: core and dumb, easy and hard, full and more, less and nice, and nonn and sign.

Offset

The offset is the index of the first term given. For some sequences, the offset is obvious. For example, if we list the sequence of square numbers as 0, 1, 4, 9, 16, 25 ..., the offset is 0; while if we list it as 1, 4, 9, 16, 25 ..., the offset is 1. The default offset is 0, and most sequences in the OEIS have offset of either 0 or 1. Sequence A073502, the magic constant for n×n magic square with prime entries (regarding 1 as a prime) with smallest row sums, is an example of a sequence with offset 3, and A072171, "Number of stars of visual magnitude n." is an example of a sequence with offset -1. Sometimes there can be disagreement over what the initial terms of the sequence are, and correspondingly what the offset should be. In the case of the lazy caterer's sequence, the maximum number of pieces you can cut a pancake into with n cuts, the OEIS gives the sequence as 1, 2, 4, 7, 11, 16, 22, 29, 37, ... A000124, with offset 0, while Mathworld gives the sequence as 2, 4, 7, 11, 16, 22, 29, 37, ... (implied offset 1). It can be argued that making no cuts to the pancake is technically a number of cuts, namely n = 0. But it can also be argued that an uncut pancake is irrelevant to the problem. Although the offset is a required field, some contributors don't bother to check if the default offset of 0 is appropriate to the sequence they are sending in. The internal format actually shows two numbers for the offset. The first is the number described above, while the second represents the index of the first entry (counting from 1) that has an absolute value greater than 1. This second value is used to speed up the process of searching for a sequence. Thus A000001, which starts 1, 1, 1, 2 with the first entry representing a(1) has 1, 4 as the internal value of the offset field.

Author(s)

The author(s) of the sequence is (are) the person(s) who submitted the sequence, even if the sequence has been known since ancient times. The name of the submitter(s) is given first name (spelled out in full), middle initial(s) (if applicable) and last name; this in contrast to the way names are written in the reference fields. The e-mail address of the submitter is also given, with the @ character replaced by "(AT)" with some exceptions such as for associate editors or if an e-mail address does not exist. For most sequences after A055000, the author field also includes the date the submitter sent in the sequence.

Extension

Names of people who extended (added more terms to) the sequence, followed by date of extension.

Searching the OEIS

The previous version of the main look-up page of the OEIS offered three ways to look up sequences, and the right radio button had to be selected. There was an advanced look-up page, but its usefulness has been integrated into the main look-up page in a major redesign of the interface in January 2006.

Enter a sequence

Enter a few terms of the sequence, separated by either spaces or commas (or both). You can enter negative signs, but they will be ignored. For example, 0, 3, 7, 13, 20, 28, 36, 43, 47, 45, 32, 0, −64, n² minus the nth Fibonacci number, is a sequence that is technically not in the OEIS, but the very similar sequence 0, −3, −7, −13, −20, −28, −36, −43, −47, −45, −32, 0, 64, is in the OEIS and will come up when one searches for its reversed signs counterpart. However, the search can be forced to match signs by using the prefix "signed:" in the search string. This is especially useful for sequences like A008836 that consist exclusively of positive and negative ones. One can enter as little as a single integer or as much as four lines of terms. Sloane recommends entering six terms, a(2) to a(7), in order to get enough results, but not too many results. There are cases where entering just one integer gives precisely one result, such as 6610199 brings up just A133207 (the strobogrammatic primes which are not palindromic). There are also cases where one can enter many terms and still not narrow the results down very much.

Enter a word

Enter a string of alphanumerical characters. Certain characters, like accented foreign letters, are not allowed. Thus, to search for sequences relating to Znám's problem, try enter it without the accents: "Znam's problem". The handling of apostrophes has been greatly improved in the 2006 redesign. The search strings "Pascal's triangle," "Pascals triangle" and "Pascal triangle" all give the desired results. To look up most polygonal numbers by word, try "n-gonal numbers" rather than "Greek prefix-gonal numbers" (e.g., "47-gonal numbers" instead of "tetracontaheptagonal numbers"). Beyond "dodecagonal numbers," word searching with the Greek prefixes might fail to yield the desired results.

Enter a sequence number

Enter the modern OEIS A number of the sequence, with the letter A and with or without zero-padding. As of 2006, the old M and N sequence numbers will yield the proper result as search strings, e.g., a search for M0422 will correctly bring up A006047, the number of entries in nth row of Pascal's triangle not divisible by 3 (M0422 in the book The Encyclopedia of Integer Sequences) and not A000422, concatenation of numbers from n down to 1.

Sloane's gap

In 2009, the OEIS database was used by an amateur mathematician to measure the "importance" of each integer number. The result shown in the plot on the right shows a clear "gap" between two distinct points clouds the "uninteresting numbers" (blue dots) and the "interesting" numbers that occur comparatively more often in series from the OEIS. It contains essentially prime numbers (red), numbers of the form a^n (green) and highly composite numbers (yellow). This phenomenon was studied by Nicolas Gauvrit, Jean-Paul Delahaye and Hector Zenil who explained it by social factors.