On-Line Encyclopedia of Integer Sequences

From Wikipedia for FEVERv2
Jump to navigation Jump to search

"OEIS" redirects here. On-Line Encyclopedia of Integer Sequences_sentence_0

For the birth defect known as OEIS complex, see Cloacal exstrophy. On-Line Encyclopedia of Integer Sequences_sentence_1

On-Line Encyclopedia of Integer Sequences_table_infobox_0

On-Line Encyclopedia of Integer SequencesOn-Line Encyclopedia of Integer Sequences_table_caption_0
Created byOn-Line Encyclopedia of Integer Sequences_header_cell_0_0_0 Neil SloaneOn-Line Encyclopedia of Integer Sequences_cell_0_0_1
URLOn-Line Encyclopedia of Integer Sequences_header_cell_0_1_0 On-Line Encyclopedia of Integer Sequences_cell_0_1_1
CommercialOn-Line Encyclopedia of Integer Sequences_header_cell_0_2_0 NoOn-Line Encyclopedia of Integer Sequences_cell_0_2_1
RegistrationOn-Line Encyclopedia of Integer Sequences_header_cell_0_3_0 OptionalOn-Line Encyclopedia of Integer Sequences_cell_0_3_1
LaunchedOn-Line Encyclopedia of Integer Sequences_header_cell_0_4_0 1996; 24 years ago (1996)On-Line Encyclopedia of Integer Sequences_cell_0_4_1

The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequences. On-Line Encyclopedia of Integer Sequences_sentence_2

It was created and maintained by Neil Sloane while a researcher at AT&T Labs. On-Line Encyclopedia of Integer Sequences_sentence_3

He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. On-Line Encyclopedia of Integer Sequences_sentence_4

Sloane is president of the OEIS Foundation. On-Line Encyclopedia of Integer Sequences_sentence_5

OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. On-Line Encyclopedia of Integer Sequences_sentence_6

As of November 2020 it contains 338526 sequences, making it the largest database of its kind. On-Line Encyclopedia of Integer Sequences_sentence_7

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. On-Line Encyclopedia of Integer Sequences_sentence_8

The database is searchable by keyword and by subsequence. On-Line Encyclopedia of Integer Sequences_sentence_9

History On-Line Encyclopedia of Integer Sequences_section_0

Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics. On-Line Encyclopedia of Integer Sequences_sentence_10

The database was at first stored on punched cards. On-Line Encyclopedia of Integer Sequences_sentence_11

He published selections from the database in book form twice: On-Line Encyclopedia of Integer Sequences_sentence_12

On-Line Encyclopedia of Integer Sequences_ordered_list_0

  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.On-Line Encyclopedia of Integer Sequences_item_0_0
  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.On-Line Encyclopedia of Integer Sequences_item_0_1

These books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. On-Line Encyclopedia of Integer Sequences_sentence_13

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). On-Line Encyclopedia of Integer Sequences_sentence_14

As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. On-Line Encyclopedia of Integer Sequences_sentence_15

The database continues to grow at a rate of some 10,000 entries a year. On-Line Encyclopedia of Integer Sequences_sentence_16

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. On-Line Encyclopedia of Integer Sequences_sentence_17

In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, , which counts the marks on the Ishango bone. On-Line Encyclopedia of Integer Sequences_sentence_18

In 2006, the user interface was overhauled and more advanced search capabilities were added. On-Line Encyclopedia of Integer Sequences_sentence_19

In 2010 an at was created to simplify the collaboration of the OEIS editors and contributors. On-Line Encyclopedia of Integer Sequences_sentence_20

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. On-Line Encyclopedia of Integer Sequences_sentence_21

A300000 was defined in February 2018, and by end of July 2020 the database contained more than 336,000 sequences. On-Line Encyclopedia of Integer Sequences_sentence_22

Non-integers On-Line Encyclopedia of Integer Sequences_section_1

Conventions On-Line Encyclopedia of Integer Sequences_section_2

The OEIS was limited to plain ASCII text until 2011, and it still uses a linear form of conventional mathematical notation (such as f(n) for functions, n for running variables, etc.). On-Line Encyclopedia of Integer Sequences_sentence_23

Greek letters are usually represented by their full names, e.g., mu for μ, phi for φ. On-Line Encyclopedia of Integer Sequences_sentence_24

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. On-Line Encyclopedia of Integer Sequences_sentence_25

Individual terms of sequences are separated by commas. On-Line Encyclopedia of Integer Sequences_sentence_26

Digit groups are not separated by commas, periods, or spaces. On-Line Encyclopedia of Integer Sequences_sentence_27

In comments, formulas, etc., a(n) represents the nth term of the sequence. On-Line Encyclopedia of Integer Sequences_sentence_28

Special meaning of zero On-Line Encyclopedia of Integer Sequences_section_3

Zero is often used to represent non-existent sequence elements. On-Line Encyclopedia of Integer Sequences_sentence_29

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." On-Line Encyclopedia of Integer Sequences_sentence_30

The value of a(1) (a 1×1 magic square) is 2; a(3) is 1480028129. On-Line Encyclopedia of Integer Sequences_sentence_31

But there is no such 2×2 magic square, so a(2) is 0. On-Line Encyclopedia of Integer Sequences_sentence_32

This special usage has a solid mathematical basis in certain counting functions. On-Line Encyclopedia of Integer Sequences_sentence_33

For example, the totient valence function Nφ(m) () 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. On-Line Encyclopedia of Integer Sequences_sentence_34

Occasionally −1 is used for this purpose instead, as in . On-Line Encyclopedia of Integer Sequences_sentence_35

Lexicographical ordering On-Line Encyclopedia of Integer Sequences_section_4

The OEIS maintains the lexicographical order of the sequences, so each sequence has a predecessor and a successor (its "context"). On-Line Encyclopedia of Integer Sequences_sentence_36

OEIS normalizes the sequences for lexicographical ordering, (usually) ignoring all initial zeros and ones, and also the sign of each element. On-Line Encyclopedia of Integer Sequences_sentence_37

Sequences of weight distribution codes often omit periodically recurring zeros. On-Line Encyclopedia of Integer Sequences_sentence_38

On-Line Encyclopedia of Integer Sequences_unordered_list_1

  • 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, ...On-Line Encyclopedia of Integer Sequences_item_1_2
  • Sequence #2: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, ...On-Line Encyclopedia of Integer Sequences_item_1_3
  • Sequence #3: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ...On-Line Encyclopedia of Integer Sequences_item_1_4
  • Sequence #4: 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, ...On-Line Encyclopedia of Integer Sequences_item_1_5
  • Sequence #5: 1, −3, −8, −3, −24, 24, −48, −3, −8, 72, −120, 24, −168, 144, ...On-Line Encyclopedia of Integer Sequences_item_1_6

whereas unnormalized lexicographic ordering would order these sequences thus: #3, #5, #4, #1, #2. On-Line Encyclopedia of Integer Sequences_sentence_39

Self-referential sequences On-Line Encyclopedia of Integer Sequences_section_5

Very early in the history of the OEIS, sequences defined in terms of the numbering of sequences in the OEIS itself were proposed. On-Line Encyclopedia of Integer Sequences_sentence_40

"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! On-Line Encyclopedia of Integer Sequences_sentence_41

", Sloane reminisced. On-Line Encyclopedia of Integer Sequences_sentence_42

One of the earliest self-referential sequences Sloane accepted into the OEIS was (later ) "a(n) = n-th term of sequence An or -1 if An has fewer than n terms". On-Line Encyclopedia of Integer Sequences_sentence_43

This sequence spurred progress on finding more terms of . On-Line Encyclopedia of Integer Sequences_sentence_44

lists the first term given in sequence An, but it needs to be updated from time to time because of changing opinions on offsets. On-Line Encyclopedia of Integer Sequences_sentence_45

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. On-Line Encyclopedia of Integer Sequences_sentence_46

This line of thought leads to the question "Does sequence An contain the number n ?" On-Line Encyclopedia of Integer Sequences_sentence_47

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". On-Line Encyclopedia of Integer Sequences_sentence_48

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's not in , the prime numbers. On-Line Encyclopedia of Integer Sequences_sentence_49

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): On-Line Encyclopedia of Integer Sequences_sentence_50

On-Line Encyclopedia of Integer Sequences_unordered_list_2

  • 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.On-Line Encyclopedia of Integer Sequences_item_2_7
  • 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 (again, by definition) it should be. This is a form of Russell's paradox. Hence it is also not possible to answer if 53169 is in A053873.On-Line Encyclopedia of Integer Sequences_item_2_8

Abridged example of a typical entry On-Line Encyclopedia of Integer Sequences_section_6

This entry, , was chosen because it contains every field an OEIS entry can have. On-Line Encyclopedia of Integer Sequences_sentence_51

Entry fields On-Line Encyclopedia of Integer Sequences_section_7

On-Line Encyclopedia of Integer Sequences_description_list_3

  • 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 multiple 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 shows, the rough correspondence holds.On-Line Encyclopedia of Integer Sequences_item_3_9

On-Line Encyclopedia of Integer Sequences_table_general_1

On-Line Encyclopedia of Integer Sequences_header_cell_1_0_0 Numbers n such that the binomial coefficient C(2n, n) is not divisible by the square of an odd prime.On-Line Encyclopedia of Integer Sequences_cell_1_0_1 Jan 1, 2001On-Line Encyclopedia of Integer Sequences_cell_1_0_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_1_0 Fibonacci(n)!.On-Line Encyclopedia of Integer Sequences_cell_1_1_1 Mar 14, 2001On-Line Encyclopedia of Integer Sequences_cell_1_1_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_2_0 Number of 3-dimensional polyominoes (or polycubes) with n cells and symmetry group of order exactly 24.On-Line Encyclopedia of Integer Sequences_cell_1_2_1 Jan 1, 2002On-Line Encyclopedia of Integer Sequences_cell_1_2_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_3_0 Smallest number such that n·a(n) is a concatenation of n consecutive integers ...On-Line Encyclopedia of Integer Sequences_cell_1_3_1 Aug 31, 2002On-Line Encyclopedia of Integer Sequences_cell_1_3_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_4_0 Continued fraction for ζ(3/2)On-Line Encyclopedia of Integer Sequences_cell_1_4_1 Jan 1, 2003On-Line Encyclopedia of Integer Sequences_cell_1_4_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_5_0 Number of permutations satisfying −k ≤ p(i) − i ≤ r and p(i) − iOn-Line Encyclopedia of Integer Sequences_cell_1_5_1 Feb 10, 2003On-Line Encyclopedia of Integer Sequences_cell_1_5_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_6_0 Length of longest contiguous block of 1s in binary expansion of nth prime.On-Line Encyclopedia of Integer Sequences_cell_1_6_1 Nov 20, 2003On-Line Encyclopedia of Integer Sequences_cell_1_6_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_7_0 Exponential convolution of A069321(n) with itself, where we set A069321(0) = 0.On-Line Encyclopedia of Integer Sequences_cell_1_7_1 Jan 1, 2004On-Line Encyclopedia of Integer Sequences_cell_1_7_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_8_0 Marks from the 22000-year-old Ishango bone from the Congo.On-Line Encyclopedia of Integer Sequences_cell_1_8_1 Nov 7, 2004On-Line Encyclopedia of Integer Sequences_cell_1_8_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_9_0 Column 1 of triangle A102230, and equals the convolution of A032349 with A032349 shift right.On-Line Encyclopedia of Integer Sequences_cell_1_9_1 Jan 1, 2005On-Line Encyclopedia of Integer Sequences_cell_1_9_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_10_0 Number of consecutive integers starting with n needed to sum to a Niven number.On-Line Encyclopedia of Integer Sequences_cell_1_10_1 Jul 8, 2005On-Line Encyclopedia of Integer Sequences_cell_1_10_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_11_0 Triangle-free positive integers.On-Line Encyclopedia of Integer Sequences_cell_1_11_1 Jan 12, 2006On-Line Encyclopedia of Integer Sequences_cell_1_11_2
On-Line Encyclopedia of Integer Sequences_header_cell_1_12_0 Möbius transform of sum of prime factors of n with multiplicity.On-Line Encyclopedia of Integer Sequences_cell_1_12_1 Jun 2, 2006On-Line Encyclopedia of Integer Sequences_cell_1_12_2

On-Line Encyclopedia of Integer Sequences_description_list_4

  • 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 four 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.On-Line Encyclopedia of Integer Sequences_item_4_10
  • 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.On-Line Encyclopedia of Integer Sequences_item_4_11
  • 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, () is named "The cubes: a(n) = n^3.".On-Line Encyclopedia of Integer Sequences_item_4_12
  • Comments: The comments field is for information about the sequence that does not 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 () and second Bessel polynomials () in a comment to A003215.On-Line Encyclopedia of Integer Sequences_item_4_13
  • References: References to printed documents (books, papers, ...).On-Line Encyclopedia of Integer Sequences_item_4_14
  • Links: Links, i.e. URLs, to online resources. These may be:On-Line Encyclopedia of Integer Sequences_item_4_15
    1. references to applicable articles in journalsOn-Line Encyclopedia of Integer Sequences_item_4_16
    2. links to the indexOn-Line Encyclopedia of Integer Sequences_item_4_17
    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 linesOn-Line Encyclopedia of Integer Sequences_item_4_18
    4. links to images in the local database directories which often provide combinatorial background related to graph theoryOn-Line Encyclopedia of Integer Sequences_item_4_19
    5. others related to computer codes, more extensive tabulations in specific research areas provided by individuals or research groupsOn-Line Encyclopedia of Integer Sequences_item_4_20

On-Line Encyclopedia of Integer Sequences_description_list_5

  • Formula: Formulae, recurrences, generating functions, etc. for the sequence.On-Line Encyclopedia of Integer Sequences_item_5_21
  • Example: Some examples of sequence member values.On-Line Encyclopedia of Integer Sequences_item_5_22
  • Maple: Maple code.On-Line Encyclopedia of Integer Sequences_item_5_23
  • Mathematica: Wolfram Language code.On-Line Encyclopedia of Integer Sequences_item_5_24
  • 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.On-Line Encyclopedia of Integer Sequences_item_5_25
  • 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.On-Line Encyclopedia of Integer Sequences_item_5_26
  • See also: Sequence cross-references originated by the original submitter are usually denoted by "Cf."On-Line Encyclopedia of Integer Sequences_item_5_27
  • 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:On-Line Encyclopedia of Integer Sequences_item_5_28

On-Line Encyclopedia of Integer Sequences_table_general_2

On-Line Encyclopedia of Integer Sequences_header_cell_2_0_0 3, 8, 3, 9, 4, 5, 2, 3, 1, 2, ...On-Line Encyclopedia of Integer Sequences_cell_2_0_1 Decimal expansion of ln(93/2).On-Line Encyclopedia of Integer Sequences_cell_2_0_2
On-Line Encyclopedia of Integer Sequences_header_cell_2_1_0 1, 1, 1, 3, 8, 3, 10, 1, 110, 3, 406, 3On-Line Encyclopedia of Integer Sequences_cell_2_1_1 First numerator and then denominator of the central

elements of the 1/3-Pascal triangle (by row).On-Line Encyclopedia of Integer Sequences_cell_2_1_2

On-Line Encyclopedia of Integer Sequences_header_cell_2_2_0 1, 3, 8, 3, 12, 24, 16, 3, 41, 36, 24, ...On-Line Encyclopedia of Integer Sequences_cell_2_2_1 Number of similar sublattices of Z of index n.On-Line Encyclopedia of Integer Sequences_cell_2_2_2
On-Line Encyclopedia of Integer Sequences_header_cell_2_3_0 1, −3, −8, −3, −24, 24, −48, −3, −8, 72, ...On-Line Encyclopedia of Integer Sequences_cell_2_3_1 Generated from Riemann zeta function...On-Line Encyclopedia of Integer Sequences_cell_2_3_2
On-Line Encyclopedia of Integer Sequences_header_cell_2_4_0 0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840,

504, 420, 5760, 3360, 2688, 1260On-Line Encyclopedia of Integer Sequences_cell_2_4_1

Decomposition of Stirling's S(n, 2) based on

associated numeric partitions.On-Line Encyclopedia of Integer Sequences_cell_2_4_2

On-Line Encyclopedia of Integer Sequences_header_cell_2_5_0 1, 1, 1, 0, −3, −8, −3, 56, 217, 64, −2951, −12672, ...On-Line Encyclopedia of Integer Sequences_cell_2_5_1 Expansion of exp(sin x).On-Line Encyclopedia of Integer Sequences_cell_2_5_2
On-Line Encyclopedia of Integer Sequences_header_cell_2_6_0 3, 8, 4, 1, 4, 9, 9, 0, 0, 7, 5, 4, 3, 5, 0, 7, 8On-Line Encyclopedia of Integer Sequences_cell_2_6_1 Decimal expansion of upper bound for the r-values

supporting stable period-3 orbits in the logistic equation.On-Line Encyclopedia of Integer Sequences_cell_2_6_2

On-Line Encyclopedia of Integer Sequences_description_list_6

  • Keyword: The OEIS has its own standard set of mostly four-letter keywords that characterize each sequence:On-Line Encyclopedia of Integer Sequences_item_6_29
    • base The results of the calculation depend on a specific positional base. For example, 2, 3, 5, 7, 11, 101, 131, 151, 181 ... 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 are defined. For example, the Mersenne primes 3, 7, 31, 127, 8191, 131071, ... 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".On-Line Encyclopedia of Integer Sequences_item_6_30
    • bref "sequence is too short to do any analysis with", for example, , Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.On-Line Encyclopedia of Integer Sequences_item_6_31
    • cofr The sequence represents a continued fraction, for example, continued fraction expansion of e () or π ().On-Line Encyclopedia of Integer Sequences_item_6_32
    • cons The sequence is a decimal expansion of a mathematical constant, like e () or π ().On-Line Encyclopedia of Integer Sequences_item_6_33
    • core A sequence that is of foundational importance to a branch of mathematics, such as the prime numbers (), the Fibonacci sequence (), etc.On-Line Encyclopedia of Integer Sequences_item_6_34
    • dead This keyword used for erroneous sequences that have appeared in papers or books, or for duplicates of existing sequences. For example, is the same as .On-Line Encyclopedia of Integer Sequences_item_6_35
    • dumb One of the more subjective keywords, for "unimportant sequences," which may or may not directly relate to mathematics, such as popular culture references, arbitrary sequences from Internet puzzles, and sequences related to numeric keypad entries. , "Mix digits of pi and e." is one example of lack of importance, and , "Price is Right wheel" (the sequence of numbers on the Showcase Showdown wheel used in the U.S. game show The Price Is Right) is an example of a non-mathematics-related sequence, kept mainly for trivia purposes.On-Line Encyclopedia of Integer Sequences_item_6_36
    • 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, ... , 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).On-Line Encyclopedia of Integer Sequences_item_6_37
    • eigen A sequence of eigenvalues.On-Line Encyclopedia of Integer Sequences_item_6_38
    • fini The sequence is finite, although it might still contain more terms than can be displayed. For example, the sequence field of shows only about a quarter of all the terms, but a comment notes that the last term is 3888.On-Line Encyclopedia of Integer Sequences_item_6_39
    • 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 , where the sequence of numerators would be . This keyword should not be used for sequences of continued fractions, cofr should be used instead for that purpose.On-Line Encyclopedia of Integer Sequences_item_6_40
    • 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 , of which there are precisely fifteen.On-Line Encyclopedia of Integer Sequences_item_6_41
    • 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?" lists the first ten known solutions.On-Line Encyclopedia of Integer Sequences_item_6_42
    • hear A sequence with a graph audio deemed to be "particularly interesting and/or beautiful".On-Line Encyclopedia of Integer Sequences_item_6_43
    • less A "less interesting sequence".On-Line Encyclopedia of Integer Sequences_item_6_44
    • look A sequence with a graph visual deemed to be "particularly interesting and/or beautiful".On-Line Encyclopedia of Integer Sequences_item_6_45
    • more More terms of the sequence are wanted. Readers can submit an extension.On-Line Encyclopedia of Integer Sequences_item_6_46
    • 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 , a(12) = a(3)a(4) = -8 × -3.On-Line Encyclopedia of Integer Sequences_item_6_47
    • 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.On-Line Encyclopedia of Integer Sequences_item_6_48
    • nice Perhaps the most subjective keyword of all, for "exceptionally nice sequences."On-Line Encyclopedia of Integer Sequences_item_6_49
    • 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., n, the cubes, which are all positive from n = 0 forwards) and those that by definition are completely nonnegative (e.g., n, the squares).On-Line Encyclopedia of Integer Sequences_item_6_50
    • obsc The sequence is considered obscure and needs a better definition.On-Line Encyclopedia of Integer Sequences_item_6_51
    • 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.On-Line Encyclopedia of Integer Sequences_item_6_52
    • tabf "An irregular (or funny-shaped) array of numbers made into a sequence by reading it row by row." For example, , "Triangle read by rows giving successive states of cellular automaton generated by "rule 62."On-Line Encyclopedia of Integer Sequences_item_6_53
    • 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, .On-Line Encyclopedia of Integer Sequences_item_6_54
    • 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.On-Line Encyclopedia of Integer Sequences_item_6_55
    • unkn "Little is known" about the sequence, not even the formula that produces it. For example, , which was presented to the Internet Oracle to ponder.On-Line Encyclopedia of Integer Sequences_item_6_56
    • walk "Counts walks (or self-avoiding paths)."On-Line Encyclopedia of Integer Sequences_item_6_57
    • 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 ... , "Number of letters in the English name of n, excluding spaces and hyphens."On-Line Encyclopedia of Integer Sequences_item_6_58
  • Some keywords are mutually exclusive, namely: core and dumb, easy and hard, full and more, less and nice, and nonn and sign.On-Line Encyclopedia of Integer Sequences_item_6_59
  • 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 , 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 , "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, ... , 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 , which starts 1, 1, 1, 2 with the first entry representing a(1) has 1, 4 as the internal value of the offset field.On-Line Encyclopedia of Integer Sequences_item_6_60
  • 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.On-Line Encyclopedia of Integer Sequences_item_6_61
  • Extension: Names of people who extended (added more terms to) the sequence, followed by date of extension.On-Line Encyclopedia of Integer Sequences_item_6_62

Sloane's gap On-Line Encyclopedia of Integer Sequences_section_8

In 2009, the OEIS database was used by Philippe Guglielmetti to measure the "importance" of each integer number. On-Line Encyclopedia of Integer Sequences_sentence_52

The result shown in the plot on the right shows a clear "gap" between two distinct point clouds the "uninteresting numbers" (blue dots) and the "interesting" numbers that occur comparatively more often in sequences from the OEIS. On-Line Encyclopedia of Integer Sequences_sentence_53

It contains essentially prime numbers (red), numbers of the form a (green) and highly composite numbers (yellow). On-Line Encyclopedia of Integer Sequences_sentence_54

This phenomenon was studied by Nicolas Gauvrit, Jean-Paul Delahaye and Hector Zenil who explained the speed of the two clouds in terms of algorithmic complexity and the gap by social factors based on an artificial preference for sequences of primes, even numbers, geometric and Fibonacci-type sequences and so on. On-Line Encyclopedia of Integer Sequences_sentence_55

Sloane's gap was featured on a Numberphile video in 2013. On-Line Encyclopedia of Integer Sequences_sentence_56

See also On-Line Encyclopedia of Integer Sequences_section_9

On-Line Encyclopedia of Integer Sequences_unordered_list_7


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/On-Line Encyclopedia of Integer Sequences.