On-Line Encyclopedia of Integer Sequences

Related subjects: Mathematics

Background Information

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The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an extensive searchable database of integer sequences, freely available on the Web.

OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, and is widely cited. It contains over 135,000 sequences, making it the largest database of its kind.

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 and by subsequence.

History

Neil Sloane started collecting integer sequences as a student in the mid-1960's to support his work in combinatorics. The database was at first stored on punch cards. He published selections from the database in book form twice:

A Handbook of Integer Sequences (1973, ISBN 0-12-648550-X), containing 2,400 sequences.
The Encyclopedia of Integer Sequences with Simon Plouffe (1995, ISBN 0-12-558630-2), containing 5,487 sequences.

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 (1995). 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.

As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998.

In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, A100000. In 2006, the user interface was overhauled and more advanced search capabilities were added.

Non-integers

Besides integer sequences strictly speaking, OEIS also catalogued 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, ${1 \over 5}, {1 \over 4}, {1 \over 3}, {2 \over 5}, {1 \over 2}, {3 \over 5}, {2 \over 3}, {3 \over 4}, {4 \over 5}$ , 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)) or continued fraction expansions (here 3, 7, 15, 1, 292, 1, ... ( A001203)).

Conventions

The OEIS is currently limited to plain ASCII text, so it 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 $N_\phi(m)$ ( 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 A072041.

Lexicographic ordering

The OEIS maintains the lexicographic order of the sequences, so each sequence has a predecessor and a successor (its "context"). OEIS normalizes the sequences for lexicographic ordering, (usually) ignoring initial zeros or 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 ${\zeta(n + 2)} \over {\zeta(n)}$ . 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, ...

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

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

Sequence #4: 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, ...

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

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

Self-referentiality

Very early in the history of the OEIS, many people suggested sequences derived from the placement of sequences in the OEIS itself. "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. For larger n that correspond to sequences that are finite and given in full (keywords "fini" and "full"), term a(n) of A091967 is undefined.

A100544 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(1) of sequence An 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 "Is n in sequence An?" and the delightfully paradoxical sequences A053873, n is in An, and A053169, n is not in An. 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. The paradox is, which sequences do 53169 and 53873 belong to? (This is a form of Russell's paradox.)

An abridged example of a typical OEIS entry

This entry, A046970, was chosen because, with the exception of a Maple program, it contains every field an OEIS entry can have.

ID Number: A046970
URL:       http://www.research.att.com/projects/OEIS?Anum=A046970
Sequence:  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
Signed:    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
Name:      Generated from Riemann Zeta function: coefficients in series
              expansion of Zeta(n+2)/Zeta(n).
Comments:  ...
           Apart from signs also Sum_{d|n} core(d)^2*mu(n/d) where core(x) is
              the squarefree part of x. - Benoit Cloitre
              (abcloitre(AT)modulonet.fr), May 31 2002
References M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,
              Dover Publications, 1965, pp. 805-811.
Links:     Wikipedia, Riemann zeta function.
Formula:   Multiplicative with a(p^e) = 1-p^2. a(n) = Sum_{d|n} mu(d)*d^2.
Example:   a(3) = -8 because the divisors of 3 are {1, 3}, and mu(1)*1^2 + mu(3)*3^2 =
              -8.
           a(4) = -3 because the divisors of 4 are {1, 2, 4}, and mu(1)*1^2 +
              mu(2)*2^2 + mu(4)*4^2 = -3
Math'ca:   muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n,
              60}] (Lopez)
Program:   (PARI) A046970(n)=sumdiv(n,d,d^2*moebius(d)) (Benoit Cloitre)
See also:  Sequence in context: A016623 A046543 A035292 this_sequence A058936
              A002017 A086179
           Adjacent sequences: A046967 A046968 A046969 this_sequence A046971
              A046972 A046973
           Cf. A027641 and A027642.
Keywords:  sign,mult
Offset:    1
Author(s): Douglas Stoll, dougstoll(AT)email.msn.com
Extension: Corrected and extended by Vladeta Jovovic (vladeta(AT)Eunet.yu),
              Jul 25 2001
           ...

Entry fields

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.

A059097	Numbers n such that the binomial coefficient C(2n,n) is not divisible by the square of an odd prime.	January 1, 2001
A060001	Fibonacci(n)!.	March 14, 2001
A066288	Number of 3-dimensional polyominoes (or polycubes) with n cells and symmetry group of order exactly 24.	January 1, 2002
A075000	Smallest number such that n*a(n) is a concatenation of n consecutive integers ...	August 31, 2002
A078470	Continued fraction for Zeta(3/2)	January 1, 2003
A080000	Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i	February 10, 2003
A090000	Length of longest contiguous block of 1's in binary expansion of n-th prime.	November 20, 2003
A091345	Exponential convolution of A069321(n) with itself, where we set A069321(0)=0.	January 1, 2004
A100000	Marks from the 22000-year-old Ishango bone from the Congo.	November 7, 2004
A102231	Column 1 of triangle A102230, and equals the convolution of A032349 with A032349 shift right.	January 1, 2005
A110030	Number of consecutive integers starting with n needed to sum to a Niven number.	July 8, 2005
A112886	Triangle-free positive integers.	January 12, 2006
A120007	Mobius transform of sum of prime factors of n	June 2, 2006

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 M plus 4 digits, zero-padded where necessary), and the 1995 Encyclopedia of Integer Sequences contained 5487 sequences, also numbered by lexicographic order (the letter N 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.

URL

The URL field gives the preferred format for the URL to link to the sequence in question, to simplify cut and paste.

Sequence

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.

Any negative signs are stripped from this field, and the values with signs are put in the Signed field.

Signed

The signed field is almost the same thing as the sequence field except that it shows negative signs. This field is only included for sequences that have negative values. Any entry with this field must have the keyword "sign".

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 Sloane points out the unexpected relationship between centered hexagonal numbers ( A003215) and second Bessel polynomials ( A001498) in a comment to A003215.

If no name is given for a comment, the comment was made by the original submitter of the sequence.

Maple, Mathematica, and other programs

Maple and Mathematica are the preferred programs for calculating sequences in the OEIS, and they both get their own field labels, "Maple" and "Mathematica." Any other program gets a generic "Program" field label and the name of the program in parentheses. The OEIS has programs in PARI, Magma, Matlab, Python and even Microsoft Excel.

If there is no name given, the program was written by the original submitter of the sequence.

Keywords

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, 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.
cons The sequence is a decimal expansion of an important mathematical constant, like e or π.
core A sequence that is of foundational importance to a branch of mathematics, such as the prime numbers, the Fibonacci sequence, 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 keyboard, 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 spheres can touch another sphere of the same size?" A001116 lists the first ten known solutions.
less A "less interesting sequence".
more The next few terms of the sequence are not known.
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. One sequence with this keyword, A025046 has been looked at by at least one other OEIS contributor who was unable to reproduce the results given by the original submitter.
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 Sloane has not edited the sequence but believes it could be worth including in the OEIS. The sequence could contain computational or typographical errors. Contributors are invited to ponder the sequence and send Sloane their edition.
unkn "Little is known" about the sequence, not even the formula that produces it. For example, A072036, which was presented to an 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., 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 of the sequence is the person who submitted the sequence, even if the sequence has been known since ancient times. The name of the submitter 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 of the submitter is also given, with the @ character replaced by "(AT)". For most sequences after A055000, the author field also includes the date the submitter sent in the sequence. But when the submitter is one of the most frequent contributors the author field just has initials; "njas" for Neil Sloane himself, for example.

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 "sign:" 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 1990661 brings up just A007597, the strobogrammatic primes). And there are also cases where one can enter many terms and still not narrow the results down all that 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 "heptaquartagonal 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.

Searching from a Web Browser

An OpenSearch compatible web browser (such as Firefox 2.0 or Internet Explorer 7) can have OEIS added to its toolbar based search list by adding a provider with the following URL: http://www.research.att.com/~njas/sequences/?q=TEST&language=english One can then directly enter the sequence, e.g. "1,2,6,24,120" (without quotes) into the search bar and search OEIS easily.

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