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Clear-cut examples of keywords

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Over the years, there have been debates on SeqFan as to whether a given sequence in the OEIS does or does not merit such and such keyword. These cases present, depending on your viewpoint, delicious ambiguities or annoying problems which require the setting of arbitrary thresholds for their resolution. Meanwhile, new contributors to the OEIS generally don't put any keywords on their submissions, uncertain as to which of them apply. Or, more rarely, they put in keywords that don't apply to the sequence they've sent in.

This page therefore provides clear-cut examples of keywords in the hopes of helping new contributors have a better idea of which keywords apply to the sequences they're submitting. They will of course occasionally come up with sequences for which the keywords are not as clear-cut as these examples.

MediaWiki:Toc: Top nonn sign cons frac cofr eigen mult tabl tabf base word walk fini full easy hard bref nice core more less dumb dead dupe uned unkn obsc allocated recycled new changed probation


When the terms of a sequence are inextricably bound up with the representation of numbers in a specific conventional, positional base (quite often base 10, somewhat less often binary), then the sequence merits keyword:base. Algebraic expressions can be written for almost any sequence. But, as a rule of thumb, if it is much easier to define a sequence by reference to digits in a given base than it is with an algebraic expression, then we're talking about a base sequence. For our example:

Example: A029976, Palindromic primes in base 8. {... , 73, 89, 97, 113, 211, 227, 251, 349, ...} (that is, {... , 111, 131, 141, 323, 343, 373, 535, ...} )

To illustrate the rule of thumb, we could try writing an algebraic expression to define this sequence.

A029976: Prime numbers \scriptstyle p \, such that \scriptstyle \prod_{i = 0}^{\lfloor \frac{\mathcal{L} - 1}{2} \rfloor} \delta_{d_i}^{d_{\mathcal{L} - i}} \,=\, 1 \,, where \scriptstyle \mathcal{L} \, is the number of base 8 digits \scriptstyle p \, has, \scriptstyle d_0 \, is the least significant digit of \scriptstyle p \, and \scriptstyle d_{\mathcal{L} - 1} \, is the most significant, and \scriptstyle \delta_n^m \, is the Kronecker delta.

Repunits were considered for the example, but the algebraic expression is quite simple: \scriptstyle \frac{b^n - 1}{b - 1} \,. Less clear-cut examples of base sequences: sequences that look at multiple base representations (e.g., from binary to base \scriptstyle n - 1 \,), unconventional positional bases or conventional non-positional bases (phinary, Roman numerals).

Remember, keyword:base is not pejorative.


Some sequences are so brief it's hard to do any analysis with them.

Example: A003135 \scriptstyle n! \, is a nontrivial product of factorials. 9, 10, 16. It is conjectured that this list is complete.


Another way to have (usually irrational) constants in the OEIS is as simple continued fractions, which are given this keyword. The sequence can be finite (if the constant is rational) or periodic (as in the case of square roots). (The sequences of numerators and denominators for the convergents are sequences of fractions and thus given keyword:frac rather than keyword:cofr).

Example: A001203, the continued fraction expansion of \scriptstyle \pi \,

3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{292 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{2 +  \ddots}}}}}}}} \,

giving the sequence of partial quotients

{3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, ...}


The OEIS is a database of integer sequences, but important irrational constants can be entered, by conversion to a sequence of integers in a number of different ways. The most common way is as a decimal expansion: the base 10 digits become terms of the sequence. Once keyword:cons is approved for a particular sequence, the string "cons" becomes a link in the Keywords field, and users can click that link to see the constant written with a decimal point and no commas between the digits, and to put the constant through Plouffe's inverter (

Example: A000796 Decimal expansion of \scriptstyle \pi \,: {3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, ...}

Keyword:cons is not limited to decimal expansions of irrational constants. See for example A027606, Euler's constant \scriptstyle e \, in duodecimal and A021016, decimal expansion of \tfrac{1}{12} \,.

The issue of offsets for keyword:cons sequences has been a confusing one for many contributors. See OEIS format for decimal representation of constants for a detailed explanation.


It would be major news if a new sequence was given this keyword. This keyword is for those core, indispensable sequences of mathematics. The prime numbers, the Fibonacci numbers, the partition numbers; besides these, there are about 170 other sequences with this keyword and most of those were added very early on in the history of the OEIS. It is a very small club: less than 0.1% of sequences in the OEIS have this keyword.


Most sequences in the OEIS will undergo a fairly continual process of evolution long after the initial submission; these sequences are alive. Papers will be written about them, more efficient algorithms will be invented, new conjectures will be posed. Other sequences will most likely never be edited again. But these are not deadwood: they must be kept in the OEIS for some reason or other. Generally dead sequences point to "live" sequences.

One common reason is that they are typos in respected and/or well-known books, hence it is reasonable to assume that at least one person will look up such a sequence in the OEIS.

Another common reason is that of unwittingly duplicated sequences early in the history of the OEIS; the sequence has been in the OEIS for such a long time that there are books and journal articles which reference the A-number of the duplicated sequence and it would therefore be confusing if the A-number was recycled for a new sequence.

Example: A185444 Erroneous version of A061574: {–163, –67, –43, –19, –11, –7, –3, –2, –1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 22, 23, 29, 33, 37, 38, ...} (the cited book is missing 1, 14 and 31).

Example: A081497 Duplicate of A028270. (This sequence has been in the OEIS since March 25, 2003, and was not recognized as a duplicate until more than a year later).


This keyword is described as being for "unimportant sequences" in the help file, but there is an element of humor in the use of the keyword. Also, it has been used for sequences that reference the OEIS itself in some way.

Example: A085808: Price is Right wheel. {15, 80, 35, 60, 20, 40, 75, 55, 95, 50, 85, 30, 65, 10, 45, 70, 25, 90, 5, 100}

Example: A111157: Numbers \scriptstyle n \, such that sequence A_n in this database does not contain a prime. {4, 7, 12, 35, 56, 66, 82, ...}


This keyword is for sequences that are easy to compute and to understand.

Example: A008592 Multiples of 10: {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ...}

You could probably rattle off a hundred terms of this sequence without having to think too much about it. A115020 is also an easy sequence, but as it requires slightly more concentration, is not as clear an example as this one. Nevertheless, since we allow ourselves the use of computers nowadays, a sequence can be considered easy if it's easy for a computer but somewhat difficult for a human without the aid of a calculator. Few people today can compute square roots by hand, yet we consider a sequence like A010488, decimal expansion of \scriptstyle \sqrt{33} \,, easy (it's 5 point something, close to 6.)


An eigensequence: a fixed sequence for some transformation - see the files transforms and transforms (2) for further information.


The rules for inclusion in A Handbook of Integer Sequences and The Encyclopedia of Integer Sequences forbade finite sequences. An exception was made for certain sequences not known for certain to be infinite, such as the Mersenne primes. The OEIS allows the inclusion of finite sequences, and they are tagged with this keyword. If practical, proof of the finiteness of the sequence should be given, or at least referenced. Generally, it is helpful to include a comment along the lines of "Last term is \scriptstyle a(x) \,=\, y \,."

Example: A056757: Cube of number of divisors is larger than the number. {2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, ...} This sequence has exactly 50967 terms, the largest one being 27935107200.

The largest prime number in the sequence is 7; obviously prime numbers only have two divisors and therefore \scriptstyle p \,>\, \tau(p)^3 \, for all \scriptstyle p \,>\, 7 \,. The largest semiprime in the sequence is 62. A proof of the finiteness of the sequence could perhaps be built along these lines.

If the sequence has a B-file attached that gives all the terms of the sequence, the B-file link should include the words "complete sequence" (see for example A001476), but this is not reflected in the keywords.


A sequences of rational numbers, not all integers, is split into two sequences, one for the numerators and one for the denominators, and both are given this keyword. The fractions are all in lowest terms, though of necessity the integers in the sequence are rendered as \scriptstyle \frac{m}{1} \,.

Example: \scriptstyle \prod_{p|n}\left(1 - \frac1p\right). \, (This sequence pertains to the totient function.)

1, \tfrac{1}{2}, \tfrac{2}{3}, \tfrac{1}{2}, \tfrac{4}{5}, \tfrac{1}{3}, \tfrac{6}{7}, \tfrac{1}{2}, \cdots \,

This is split into A076512 (numerators: {1, 1, 2, 1, 4, 1, 6, 1, 2, 2, 10, 1, 12, 3, 8, 1, 16, 1, ...}) and A109395 (denominators: {1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 15, 2, 17, 3, ...})

For sequences of unit fractions, only the sequence of denominators need be added; the numerators are A000012.


Some finite sequences are so short they can be given in full well within the usual four lines that is the maximum for most infinite sequences. It almost goes without saying that if a sequence has keyword:full, it should also have keyword:fini. The advice given above suggesting a comment about the last term does not apply here.

Example: A003173: Heegner numbers, imaginary quadratic fields with unique factorization (or class number 1.) {1, 2, 3, 7, 11, 19, 43, 67, 163}. That's it. That's all of them.

As far as determining if this keyword applies is concerned, it does NOTmatter if the B-file gives all terms, the sequence does not merit this keyword. As a corollary, if a sequence has this keyword, the B-file is superfluous: you can just click the "list" link to see the sequence presented as two columns (n and a(n)). On February 12, 2013, Neil Sloane decided finite sequences that are given in full in the B-file do merit this keyword. We are currently in the process of identifying sequence entries affected by this change. But for the purpose of giving a clear-cut example of this keyword, a finite sequence short enough to be given in full without recourse to a B-file remains a better example than one that does.


Perhaps this is the keyword over which there is the greatest amount of disagreement. It is agreed that many infinite sequences get harder to compute as the indices get large enough: for example, what is the ten googolth semiprime? But this alone is not enough to make a sequence hard; the keyword is by no means a measure of the growth of a sequence. We could obtain the ten millionth semiprime without too much effort. There are sequences in the OEIS for which we can't even get the tenth term. The formula for such a sequence could turn out to be quite simple, but since we don't know it, and not for lack of trying, it is just as unfathomable as if it were a difficult formula.

Therefore, for our clear-cut example, we should choose a sequence for which the difficulty is not just computational but also theoretical.

Example: A001220: Wieferich primes: primes \scriptstyle p \, with the property that \scriptstyle p^2 \, divides \scriptstyle 2^{p - 1} - 1 \,. Only two are known: 1093 and 3511. Brute force number crunching has shown there are no more up to \scriptstyle 6.7 \times 10^{15} \,, and the third Wieferich prime may be discovered by a computer, but such a discovery would most likely be no help in formulating a theory to find the fourth Wieferich prime nor prove that there are no more if that is the case. Nor would it be any help in further searching by ruling out certain bigger primes from the search. Such a discovery may answer some of the questions there are about Wieferich primes but it would also raise new questions.

Keyword:hard often goes hand in hand with keyword:more. It has been argued that keyword:more should practically be understood from keyword:hard without having to be stated.

See User:Charles R Greathouse IV/Keywords/easy and hard for a much more in-depth discussion of these two keywords.


This is a keyword for sequences that are worth listening to using the OEIS MIDI Player. Like keyword:cons, the word "hear" in the Keywords field becomes a clickable hyperlink that takes one to the OEIS MIDI Player page with default settings for the sequence at hand.

Example: A144488: Ludwig van Beethoven's Bagatelle No. 25, "Für Elise".

The keyword was introduced January 26, 2014.


This doesn't mean that we want fewer terms of the sequence, but rather that we want to lessen the ranking of the sequence in search results. Some sequences are not that interesting, but it is believed that there is some small but significant possibility that they will be looked up by others than those who submitted them. By the other side of the coin, we don't want such sequences to be the very first result in a search that also yields other, potentially more interesting, results. (Of course there are also sequences that are so uninteresting that they will be rejected altogether).

Example: A144572, Primes of the form nonprime(prime(n)) + 1 (the first nonprime under this definition is 0) {2, 5, 11, 17, 41, 59, 73, 83, 97, ...}

If someone looks up "2, 5, 11, 17," odds are they are more interested in odd-indexed prime numbers (A031368), or maybe the first prime between two consecutive squares (A007491) than they are in primes of the form nonprime(prime(n)) + 1. As of 2011, this search brings up seven pages of results, and primes of the form nonprime(prime(n)) + 1 don't appear until the seventh page. However, if they search for "2, 5, 11, 17, 41, 59," then this could be the sequence they're looking for.


This is a keyword for sequences that are particularly worthwhile to look as a graph. Like keyword:cons, the word "hear" in the Keywords field becomes a clickable hyperlink that takes one to a page showing a pin plot and scatter plot of the sequence at hand.


The keyword was introduced January 26, 2014.


For one reason or another, some sequences in the OEIS don't have as many terms as we would like. In the case of a sequence that also has keyword:hard, the difficulty is quite obvious: we know what we're looking for but we don't where to look, nor in some cases whether it exists to be found. Or it could be the case that the original contributor computed as much as he could by hand and no one has gotten around to using a computer to extend the sequence.

Example: A123692 Primes p such that p2 divides 5p − 1 − 1. 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801. These numbers get large fairly quick, but presumably extending this sequence is just a matter of letting a computer search run long enough. In this example, the next term, if it exists, is larger than 1014.


If in contributing a new sequence, you find yourself writing a comment such as "Volkov proved this sequence is multiplicative in 2005," then you should also add keyword:mult. As you already know, a function \scriptstyle f(n) \, is multiplicative if \scriptstyle f(ab) \,=\, f(a)f(b) \, when \scriptstyle \gcd(a, b) \,=\, 1 \,, and fully or completely multiplicative when the condition of coprimality is unnecessary. For the purpose of the keyword, there is no need to distinguish between multiplicative and fully multiplicative, but for the latter a comment to that effect may sometimes be appropriate.

Example: A184997: Number of distinct remainders that are possible when a safe prime \scriptstyle p \, is divided by \scriptstyle n \, (for \scriptstyle p \,>\, 2n + 1 \,). {1, 1, 1, 1, 3, 1, 5, 2, 3, 3, 9, 1, 11, 5, 3, 4, 15, 3, 17, ...} One of the comments in the entry references the Chinese remainder theorem in saying the sequence is multiplicative.


For "an exceptionally nice sequence." Generally, you should not assign this keyword to new sequences you submit (unless of course there was a discussion prior somewhere, like on SeqFan, where everyone agreed it merits the keyword). Alternatively, you can suggest to the editors that the sequence should have the keyword.

Example: A167408, orderly numbers: a number n is orderly if there exists some number k > τ(n) such that the set of the divisors of n is congruent to the set \{1, 2, \cdots, \tau(n)\} \mod k. 1, 2, 5, 7, 8, 9, 11, 12, 13, 17, 19, 20, 23, 27, 29, 31, 37, 38, 41, 43, 47, ...

There was great surprise on SeqFan that no one before Andrew Weimholt had thought of this sequence, and then unanimous agreement that it is a nice sequence.


This keyword is for sequences containing no negative terms, at least not within term visibility. The system will automatically assign it if applicable, but it doesn't hurt to understand its raison d'être.

Example: A000290, \scriptstyle n^2 \,, with \scriptstyle n \,\in\, \mathbb{Z} \,: {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...}

Even if we allow negative \scriptstyle n,\, n^2 \, is still positive (or 0 in the case of 0).

Remember: the computer knows the signs of only the terms you enter; it can't figure out the theoretical underpinnings of the sequence, much less determine if it contains negative terms beyond those you enter. If necessary, you can delete "nonn" and type in "sign" instead.


With keyword:unkn, no description is known for the sequence. With keyword:obsc, a description is known but it is not the most enlightening description.

Example: A086267, a(n) = 3 + (H(n) \mod 6) + \lfloor r \rfloor where H(n) is the nth number in the Hofstadter Q-sequence and r = \frac{H(n) -2H(n + 1) + H(n + 2) - 4}{H(n)}. The description, though convoluted and seemingly unmotivated, is at least clear enough that others can reproduce the original contributor's numbers.


This keyword is generally for sequences containing some negative terms. (If the sequence is all negative terms, then you could just multiply all terms by –1 and amend the definition accordingly; getting rid of all those minus signs might allow you to put in maybe a couple dozen more terms without having a B-file.)

Example: A100700 n-th Fibonacci number minus n-th prime number: {–1, –2, –3, –4, –6, –5, –4, 2, 11, 26, 58, 107, ...}

As you may have surmised from the explanation of keyword:nonn, the system will automatically assign keyword:sign if you enter minus signs into the Data field. But if the sequence contains negative terms only beyond term visibility, then you have to explicitly assign keyword:sign. Much less clear-cut are those cases where sequences are conjectured to contain negative terms; in such a case a comment should be entered to that effect.


What if instead of a smooth triangle, your arrangement of numbers has a very jagged edge due to wildly changing row lengths? That's what keyword:tabf is for. In the triangle of prime factors (with multiplicity), prime-numbered rows will of course have just one column, while multiples of cubes of primes will have three columns, etc., and these occur at seemingly chaotic intervals:

 2 2
 2 3
 2 2 2
 3 3
 2 5
 2 2 3
 2 7
 3 5
 2 2 2 2
 2 3 3
 2 2 5

Even though the rows are uneven, we can still this read this by rows and enter it into the OEIS, though perhaps in a case such as this case we might really wish we could avail ourselves to semicolons. At least the fact that the numbers are in ascending order in each row helps make sense of where one row ends and the next begins. Thus:

A027746, Triangle in which first row is 1, each \scriptstyle n \,th row afterwards gives prime factors of \scriptstyle n \, (with multiplicity.)

{1, 2, 3, 2, 2, 5, 2, 3, 7, 2, 2, 2, 3, 3, 2, 5, 11, 2, 2, 3, 13, 2, 7, 3, 5, 2, 2, 2, 2, 17, 2, 3, 3, 19, 2, 2, 5, ...}


Perhaps for as long as man has written down numbers, man has arranged numbers into geometrical shapes, like triangles and squares. Pascal's triangle and the magic square of order 3 were known to the ancient Chinese long before Pascal and Dürer. In the OEIS, number triangles like Pascal's triangle are made into sequences by reading them row by row. For our example, we will use Lozanić's triangle, which is closely related to Pascal's:

Lozanić's triangle
Row sums
\sum_{i = 0}^n T(n, i)
n \, = 0 1 1
1 1 1 2
2 1 1 1 3
3 1 2 2 1 6
4 1 2 4 2 1 10
5 1 3 6 6 3 1 20
6 1 3 9 10 9 3 1 36
7 1 4 12 19 19 12 4 1 72
8 1 4 16 28 38 28 16 4 1 136
9 1 5 20 44 66 66 44 20 5 1 272
10 1 5 25 60 110 126 110 60 25 5 1 528
11 1 6 30 85 170 236 236 170 85 30 6 1 1056

(Note that the sequence of row sums of a number triangle does not get keyword:tabl unless it is somehow itself another triangle).

In unformatted plaintext we might render this triangle as:

{1; 1, 1; 1, 1, 1; 1, 2, 2, 1; 1, 2, 4, 2, 1; 1, 3, 6, 6, 3, 1; 1, 3, 9, 10, 9, 3, 1; 1, ...}

but semicolons are not allowed in the Data field of sequences. This is not really a problem unless we wanted to start at a row other than the first (the row with just one element, whether it be numbered row 0 or row 1). Thus in A034851 we see that the triangle is given as {1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 10, 9, 3, 1, ...}.

Once keyword:tabl is approved for a sequence, the string "tabl" in the Keywords field becomes a clickable link that shows the sequence formatted as a triangular array, a square array and an upper right triangle. Before this feature was instituted, some contributors would use the Example field to show the triangle in preformatted plaintext.


Some sequences concern walks through a lattice of specified shape, size and/or dimensions. The mental picture of the walk is most easily realized two-dimensionally, with the lattice being a floor to walk on. The most important rule is that one can go from one lattice point (or vertex) to another in one step only if they are connected by a lattice line (or edge). So, for example, in a square lattice with the points labelled by Gaussian integers, one can't go from 0 to \scriptstyle -1 + i \, in a single step, even though the distance of \scriptstyle \sqrt{2} \, is not that much more than the distance of 1 to go to either \scriptstyle -1 \, or \scriptstyle i \, before \scriptstyle -1 + i \,. A walk may have additional rules, such as the self-avoiding walk: once you step off a point, you may not step on it again.

80 steps through a finite square lattice.


Sequences pertaining to paths and trails should also have this keyword.


In order to avoid an insurmountable backlog under the old system, some entries were added to the OEIS unedited. The sequences were interesting, but the presentation very lacking. Over time, these have been edited some time after being first added, and keyword:uned is accordingly removed. There remain quite a few unedited entries in the OEIS today.

But there is no good reason why this keyword should be given to any new sequences. Even after the first time you signal that your new entry is ready to be looked at by the Editors, there is a window of time to change your mind and tweak the entry. Then, once the Editors start looking at it, they may send it back to you for further adjustments if necessary. Under our new system, this is all managed very smoothly, and the closest we come to having a serious backlog is after holidays.


When people are stumped over how a particular sequence of numbers is defined, they turn to the OEIS. But what happens when the Editors of the OEIS don't know either? The sequence is added to the OEIS but given this keyword, in the hopes that someone will come along who does know the answer. For such a sequence, the original contributor should add as much information in the entry as is known to him: Where was the sequence found? Was an incorrect definition given for it?

Example: A058897: An unknown sequence: several people have asked about this, so I have added it to the database. {6, 39, 78, 95, 82, 4, ...} (presumably the sequence is infinite). As far as we know, the first time this sequence was ever mentioned was in a 2001 post at


What interesting correlations are there between numbers and the words for numbers in a given human language (such as Italian or Japanese)? Some sequences look at these relationships, so they get this keyword.

Example: A001166, smallest natural number requiring n letters in English. 1 (one), 4 (four), 3 (three), 11 (eleven), 15 (fifteen), 13 (thirteen), 17 (seventeen), 24 (twenty-four), 23 (twenty-three), 73 (seventy-three), etc.

Comments are in order when regional variances affect the terms of the sequence. Depending on where in the French-speaking world you go, 80 is "quatre-vingts," "huitante" or "octante," to give just one example.

Temporary or obsolete keywords

For completeness, we now consider these keywords that are meant for temporary use (keyword:more could theoretically fall in this category, but in some particularly thorny cases could very well be permanent.) Most of these are assigned automatically by the system.


As soon as a contributor clicks on "Contribute a new sequence to the OEIS," the system finds the next available A-number and gives it this keyword. The name of the sequence is then changed to "Allocated for [contributor's name]." Other keywords will be bandied about during the initial draft edits phase, but keyword:allocated will remain in effect until those edits are approved.


Older sequences that have been recently changed get keyword:changed (also new sequences edited soon after initial approval). One old sequence that can be counted on to have this keyword a lot of the time is A000040.


This keyword is included here only for historical reasons. It was meant for duplicated sequences, but generally those were given keyword:dead instead. Thus keyword:dupe became deprecated some time after 2000, and was only recently abolished completely. The distinction between erroneous versions and duplicates, as you can see from the examples for keyword:dead, is maintained in the name of the sequence.


Once a new sequence is approved, the system automatically gives it keyword:new, which remains in effect for about a week or so.


Not presently in use. Introduced for sequences which were only tentatively approved.


When the editors agree that a new proposed sequence is not worth adding to the OEIS, one editor blanks the entry leaving only the keyword line with keyword:recycled. The A-number then becomes available for allocation for another new sequence.

See also

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