This is the absolute catalog number of the sequence.
Some sequences also have a 4-digit M-number, such as M1459, which is the number they carried in "The Encyclopedia of Integer Sequences" by N. J. A. Sloane and S. Plouffe, Academic Press, San Diego, CA, 1995.
Some older sequences also have a 4-digit N-number, such as N0577, which is the number they carried in "A Handbook of Integer Sequences", by N. J. A. Sloane, Academic Press, NY, 1973.
Name
This gives a brief description or definition of the sequence. Example:
The even numbers.
In the description, a(n) denotes the n-th term of the sequence and n is a typical subscript.
Example: a(n) = a(n-1) + a(n-3).
In some cases however the definition will describe a typical term k in the sequence.
Example: Numbers k such that k and k+1 have the same number of divisors.
Avoid vanity: do not name the sequence after yourself (or your family members, your friends, etc.).
Prefer concise mathematical definitions when possible.
When submitting a sequence, once a name has been entered, it might be necessary to make small changes, but for a complete change, e.g., if the original sequence was aborted or withdrawn, do not use the same A-number; please ask for recycling and request a new A-number.
Data
This field gives the beginning (at least 4 terms) of the sequence. Example:
Ideally this field should give enough terms to fill about three lines on the screen—maybe 260 characters including [decimal] digits, signs, commas and whitespaces (spaces and newlines; DO NOT USE tabs). (In the above example no more terms were known when the sequence was created.)
The sequence may contain negative integers—if the sequence is known to contain negative integers it should be labeled with the Keyword "sign" (i.e., signed sequence); if the sequence is known NOT to contain negative integers it should be labeled with the Keyword "nonn" (i.e., nonnegative sequence).
When entering a sequence, you may separate the terms by commas or spaces.
The entries must be integers—represented with base-10 digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
If the terms are fractions, then the numerators and denominators should be entered as separate sequences, labeled with the Keyword "frac", and with cross-references connecting the two sequences.
Only sequences that are well-defined and of general interest should be submitted.
If every second term is zero (A, 0, B, 0, C, 0, D, 0, E, ...) then normally we omit the zeros and list the sequence as A, B, C, D, E, ...
Offset
The index of the first term of the sequence.
For lists, the offset should generally be 1, but sometimes 0 is allowed. See examples.
For arrays and triangles, the offset should be the index of the first row. If the triangle is regular (tabl), then the first index's offset usually is also the second index's offset. If this is not the case, the offset of the second index should be given in the name.
There is a second part to the offset after a comma, which is the 1-based index of the first term greater than 1 in absolute value. If all terms are -1, 0, or 1, the second part should be 1. You do not need to add this part; the server will calculate it for you.
Do not repeat the offsets or ranges they imply in the formulas or comments. Only restrictions of the range of validity for a certain formula need to be indicated.
Comments
Relevant information that would make the sequence name much too long and interesting side facts that don't fit in the other fields.
Generally, comments are placed in chronological order, with newer comments placed at the bottom of the field. However (at the discretion of the Editors-in-Chief), the most important comments will be placed first, even if they are out of chronological order.
With the exception of comments made by the author at the time the sequence was first submitted, all comments must identify the commenter and the date the comment was made.
Therefore there is no need to write in the Extensions field "Added a comment."
The Comments section is a good place to explain the motivation for a sequence, especially when there are no links.
There are several ways to sign a comment (and all comments not part of the original submission should be signed and dated) (see below):
The preferred format for signing a short comment is "If n is divisible by 3, so is a(n). - _Alonso del Arte_, Dec 01 2011". That is, the comment is followed by a space, a dash, a space, then the name (in the form of a link), then a comma, a space, and the date. If you type four tilde characters (~~~~), the system will automatically replace them with your username (in the form of a link), a comma, a space, and the date, all in the proper format.
Short comments can also be written as "Comment from _Alonso del Arte_, Dec 01 2011: If n is divisible by 3, so is a(n)."
Another possible format for short comments is "From _Alonso del Arte_, Dec 01 2011: If n is divisible by 3, so is a(n)."
For a comment that extends over several paragraphs, the preferred format is: "Comment from _Alonso del Arte_, Dec 01 2011: (Start)" followed (starting on a new line) by the comments, then "(End)".
If a comment contains an error, it may be corrected in place. In that case the author of the correction adds his or her signature as in the following example: "a(n) = a(n-1) + 2*(n-1). - _Henry Bottomley_, Oct 02 2000 [Corrected by _N. J. A. Sloane_, Jul 18 2010]"
References
References to books, journal papers, and other material not found online.
There are often multiple mentions of an article, in both the References and Links sections. This is perfectly normal and often desirable, for many reasons. Different versions of a paper may have appeared on the arXiv or a home page and also in a journal; a preprint version may have been sent to the OEIS and later changed in the course of publication. References to the OEIS may have appeared in one version and then dropped, or vice versa. Formulas may have changed. Please do NOT delete references or links just because they appear to you to be duplicates!
Entries should be alphabetized by the last name of the first author listed. For example, there is a book by John Gold and a paper co-authored by Yang Zhang and Sigfrid Aronson as references for a given sequence. The Gold book goes first, then the Zhang & Aronson paper. (Most math papers list authors alphabetically, but if they use a different ordering use it.)
Links
Links to journal papers, preprints, illustrations, web pages, and other material:
If a paper is available from more than one location, or in several versions, include all the links (within reason)
Indicate broken links by "[broken link]" or replace the URL with an archived version
Use stable URLs when available
Entries should be alphabetized by the last name of the first author, except that
When the author is unknown, write "Author?"; when the title is unknown, write "Title?"
When the author is a named group, use the name of the group: MathOverflow, N. Bourbaki, Wikipedia, etc.
When a single source has many links they may be combined. (E.g., several MathWorld articles may be combined into a single line if the Links section grows too large.)
Links containing hidden programs:
Links may not be used to trigger the execution of programs whose code is contained in the link in compressed and possibly encrypted form. Instead, it is recommended that these programs be uploaded to the OEIS as external files.
Formula
Formulas for calculating the n-th term of the sequence, generating functions, asymptotics, and so forth.
See section #Spelling and notation for preferred notation of mathematical formulae. In decimal expansions a formula Equals formula is about the constant, not about the decimal digits a(n).
Example
An example of how to find or interpret terms when it is not obvious.
In the special case of a constant, give the first few digits of the decimal expansion (usually to the width of the line).
In the special case of a table, give the first few rows (or the top-left corner if a rectangular array).
Don't give an example that just repeats the definition. The idea is to choose 1 to 3 simple cases, and show your work.
Maple
A Maple program.
Please sign and date your contribution in a comment, using Maple's end-of-line comment syntax, " # ~~~~ ". Or you can start with " # Maple program from ~~~~ ".
Programs should be self-contained except in rare cases.
If a program is extremely long, consider submitting it as a link and mentioning it in this field.
Please sign and date your contribution in a comment, using Mathematica's comment syntax, " (* ~~~~ *) ". Or you can start with " (* Mathematica program from ~~~~ *) ".
Programs should be self-contained except in rare cases.
If a program is extremely long, consider submitting it as a link and mentioning it in this field.
Code in the Wolfram Language belongs in this field even if it is executed via Wolfram Alpha.
Programs written specifically in the Wolfram Alpha language are strongly discouraged in the OEIS. The reason is that Wolfram Alpha also accepts inputs that are not in the formal Mathematica language. One can ask Wolfram Alpha something like: "sum the squares from 1 to n". The problem is that the interpretation of a query expressed in natural language is difficult, sometimes produces unwanted results, and even when it gives the expected result there is no guarantee that next year the same English sentence will be interpreted in the same way. Since the OEIS is a scientific database, and reproducibility is essential, programs in Wolfram Alpha are discouraged.
Programs
A program in some language other than Maple or Mathematica.
Please sign and date your contribution in a comment.
Programs should be self-contained and copy-pastable, except in rare cases.
If a program is extremely long, consider uploading it as a file to be stored with the sequence as a link. (You will see a facility for doing this when editing the LINKS section.) Mention it in this field like so:
(PARI) \\ See Smith link.
using the appropriate symbol for comments
Also give the name of the programming language in parentheses at the beginning, like (PARI) or (Perl)
Entries are generally chronological, but keep entries written in a given programming language together.
Code should be signed unless it was written by the sequence author as part of the original submission. To allow the program to run in case the signature is copied, the signature should be in a comment. The preferred formats for such comments (in languages frequently used in the OEIS), where the block ~~~~ will be automatically replaced by the user's name and the current date, are:
It is no longer necessary to replace leading spaces with dots, for languages that require indentation: as soon as a line starts with two spaces or more, they will be preserved.
Note that code in the Wolfram Language belongs in the Mathematica section, not here, regardless of whether it is executed in Mathematica or Wolfram Alpha, and that code in the Maple language belongs in the Maple section.
Cross-references
Cross-references to related sequences. This should be a comma-separated list of sequence numbers without repetition, with a space after each comma.
The author's name and the date of initial submission. Sequences with multiple authors should have multiple names here. This field generally does not change once the sequence is first approved.
Older sequences may also include the author's email address; when possible, the author's name should be linked instead.
This field generally represents the person submitting the sequence, even if the sequence was known earlier. For example A000040, the prime numbers, were known long before they were entered in the OEIS, but the author is still listed as N. J. A. Sloane.
Extensions
This field is to claim credit for additions to the entry that can't be properly acknowledged in other fields. The most common use is to acknowledge more terms for sequences that had only a few previously, e.g., "a(10)-a(24) from _Jan Schuster_, Mar 14 2015".
Comments, programs (including Maple and Mathematica), and formulas should be signed (see more about signing above and in QandA) and thus do not need to be mentioned in this field.
Cross-references, links, references, and (usually) examples are neither acknowledged in this field nor signed in place.
During the approval process, you do not need to add anything in this field when making changes to the sequence you are creating.
Status
A non-editable field that shows the status of the sequence: approved, editing, reviewed, or proposed.
When a sequence is edited, the status is automatically changed to "editing"
When the status is changed from "editing" to "proposed", the sequence is shown to the Editorial Board for review and approval
Change this by clicking "These changes are ready for review by an OEIS Editor" at the bottom of the sequence draft
A typical entry
Here is an (abbreviated) example showing the different types of lines in an entry in the OEIS:
The first line you see has two or three parts:
A000002 Kolakoski sequence: a(n) is length of n-th run; a(1) = 1; sequence consists just of 1's and 2's. ID M0190 N0070
(history; published version; edit)
No proposed changes to A000002.
NAME
Kolakoski sequence: a(n) is length of n-th run; a(1) = 1; sequence consists just of 1's and 2's.
DATA
1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2
OFFSET
1,2
COMMENTS
It is an unsolved problem to show that the density of 1's is equal to 1/2.
The sequence is cubefree and all square subwords have lengths which are one of 2, 4, 6, 18 and 54.
This is a fractal sequence: replace each run with its length and recover the original sequence. - Kerry Mitchell, Dec 08 2005
...
REFERENCES
J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 337.
E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
F. M. Dekking, On the structure of self-generating sequences, Seminar on Number Theory, 1980-1981 (Talence, 1980-1981), Exp. No. 31, 6 pp., Univ. Bordeaux I, Talence, 1981. Math. Rev. 83e:10075.
...
LINKS
N. J. A. Sloane, <a href="/A000002/b000002.txt">Table of n, a(n) for n = 1..10502</a>
J.-P. Allouche, M. Baake, J. Cassaigns and D. Damanik, <a href="http://www.lri.fr/~allouche/">Palindrome complexity</a>
Michael Baake and Bernd Sing, <a href="http://arXiv.org/abs/math.MG/0206098">Kolakoski-(3,1) is a (deformed) model set</a>
...
FORMULA
These two formulae define completely the sequence: a(1)=1, a(2)=2, a(a(1)+a(2)+...+a(k))=(3+(-1)^k)/2 and a(a(1)+a(2)+...+a(k)+1)=(3-(-1)^k)/2. - Benoit Cloitre, Oct 06 2003
a(n+2)*a(n+1)*a(n)/2 = a(n+2)+a(n+1)+a(n)-3 (this formula doesn't define the sequence, it is just a consequence of definition). - Benoit Cloitre, Nov 17 2003
a(n+1)=3-a(n)+(a(n)-a(n-1))*(a(b(n))-1), where b(n) is the sequence A156253. - Jean-Marc Fedou and Gabriele Fici, Mar 18 2010
EXAMPLE
Start with a(1) = 1, a(2) = 2. The rule says that the first run (which is a single 1) has length 1, which it does and the second run (which starts with the 2) has length 2, so the third term must be a 2 also and the fourth term can't be a 2, so must be a 1. So we have a(3) = 2, a(4) = 1. Since a(3) = 2, the third run has length 2, so we deduce a(5) = 1, a(6) = 2. And so on. The correction I made was to change a(4) to a(5) and a(5) to a(6). - Labos Elemer, corrected by Graeme McRae
MAPLE
M := 100; s := [ 1, 2, 2 ]; for n from 3 to M do for i from 1 to s[n] do s := [ op(s), 1+((n-1)mod 2) ]; od: od: s; A000002 := n->s[n];
MATHEMATICA
a[steps_] := Module[{a = {1, 2, 2}}, Do[a = Append[a, 1 + Mod[(n - 1), 2]], {n, 3, steps}, {i, a[[n]]}]; a]
PROG
(PARI) a=[ 1, 2, 2 ]; for(n=3, 80, for(i=1, a[n], a=concat(a, 1+((n-1)%2)))); a
(PARI) a(n)=local(an, m); if(n<1, 0, an=[1, 2, 2]; m=3; while(length(an)<n, an=concat(an, vector(an[m], i, (m-1)%2+1)); m++); an[n])
CROSSREFS
Cf. A064353, A001462, A001083, A006928, A042942, A069864, A010060, A078880.
Cf. A079729, A079730, A078929, A171899.
KEYWORD
nonn,core,easy,nice,new
AUTHOR
N. J. A. Sloane
EXTENSIONS
arXiv URL replaced with a non-cached version by R. J. Mathar, Oct 07 2009
STATUS
approved
How many terms do we need?
Normally an entry in the OEIS will give enough terms to fill three lines on the screen, about 260 characters including spaces and commas. (Enough to fill three punched cards, in the old days.)
260 characters is just a guide. If you want to give more, go ahead. We are not traffic cops with our radar guns set to flash red at 261 characters. There is ample space. Try to stay under 500 chars, though.
If more terms are available, then the entry may have a b-file, giving the first 1000 or 10000 or 20000 terms, or more in exceptional cases. The most common size is 10000 terms. The Editors-in-Chief have the final say about what size of b-file is appropriate.
If the entry already has about three lines' worth of terms, there is usually no point in adding more terms to the DATA lines -- instead, create a b-file.
Generally, the minimum number of terms required is 4.
Signing your name when you contribute to an existing sequence
You should sign (i.e., add your name & date) when you add something to an existing entry. This is done by appending - ~~~~ at the end of the line, which will take care of correctly formatting your name (including the link) and the date. The same information is available in the History section, but it is the standard convention in the OEIS and useful for the reader who does not want to search the history of revisions.
If you are adding something that extends over several paragraphs, then rather than putting your name on every line, put Comment from _N. J. A. Sloane_, Sep 24 2007: (Start) at the beginning and (End) at the end.
Examples
T(k,n) = n(k-2)((k-2)n^2+1+2n)/2. - _R. J. Mathar_, Jun 12 2008
Comment from _Paul D. Hanna_, Jun 14 2009: (Start)
More generally, if G(x) = exp(p*x*exp(q*x*G(x))),
where G(x)^m = Sum_{n>=0} g(n,m)*x^n/n!,
then g(n,m) = Sum_{k=0..n} C(n,k)*p^k*q^(n-k)*m*(n-k+m)^(k-1)*k^(n-k).
(End)
T(k,n) = n(k-2)((k-2)n^2+1+2n)/2. [_R. J. Mathar_, Jun 12 2008]
Email
Don't include your email address
Since January 2011, it has been the policy of the OEIS not to display email addresses of contributors.
Instead, sign your name by typing four tildes (~~~~) at the end of a line to link to your User Page. This should be automatically converted to your name and date at the Preview stage.
In comments, examples and formulae, this signature is typically preceded by " - " (space, dash, space), separating it from the (mandatory) period (".") at the end of the comment. (Example: "This is a comment. - ~~~~".) But " [~~~~] " is also acceptable (the signature is inserted between square brackets). In programs, the name and date is either put inside comment delimiters (e.g., "(*...*)" in Mathematica), or (preferably) after a comment-up-to-end-of-line introducing character or characters, if available ("#" in Maple, "\\" in PARI/GP, "//" in C; see above for other programming languages; this allows users to copy/paste the code and a possibly incomplete part of the signature without generating errors when the code is executed).
Provided this person has a user page on this Wiki (and ideally every contributor should have such a page), first go to the OEIS Wiki and log in, then go to the user's page (enter, for example, User:John Doe, in the Wiki search box). Then there should be a link in the Toolbox on the left saying "E-mail this user". Of course, there are many possible reasons why a contributor, even one with a user page, may not have a current email address.
You may also leave a message for the user on her or his User Talk page (but again you must first log in to the OEIS Wiki — the Wiki requires a separate login from the main OEIS database).
Spelling and notation
The following are the correct spellings for some words and symbols that are commonly mistyped in the OEIS, also the preferred versions of various useful symbols:
> (not grth)
>= (not ≥ or \ge or \geq)
<= (not ≤ or \le or \leq)
!=or<> for "not equal" (not ≠, .NE., \ne, \neq), e.g., x != y or x <> y.
For / write a/(bc), not a/b*c or a/b/c which are ambiguous (see operator precedence). For example, don't say 1/6x, say x/6 or (1/6)x. Don't say 1/6 Sum ..., say (1/6) Sum ... . Don't say 1/(1-x)/(1-x^2), say 1/((1-x)(1-x^2)).
|S|, #S, or card(S) are all acceptable for the cardinality of a set S
9-gon, not nonagon: numerals are used for figurate numbers above 8
a(n) for n-th term in sequence (not a[n] or a_n)
a(n) denotes a single term in the sequence. To refer to the whole sequence, use any of {a(n)}, [a(n)], or (a(n)). Of course curly brackets as in P = {2,3,5,7,11,...} usually refer to a set. When you need to distinguish between a sequence and the set of its terms, you need to spell out what you are doing in words. Say something like "Here [a(n): n >= 0] is the sequence, and {a(n)} is the set of its terms."
behavior (not behaviour - the OEIS uses US spelling)
billion, trillion, ... (10^9, 10^12, ...) (American system, in which the prefix stands for n in 10^(3+3n))
binomial(n,k) or C(n,k) for binomial coefficients; the former is preferred but the latter is acceptable in formulas if there are quite a few coefficients. Capitalize "Binomial" only at the beginning of sentence
cancellation (not cancelation) is the correct spelling
ceiling (not ceil; not Ceiling, unless at the beginning of a sentence)
color (not colour - the OEIS uses US spelling)
cos(x) (not Cos[x])
cubefree (not cube-free, not cube free)
dependent (not dependant)
dissectable (not dissectible)
e for 2.718281828459... (not E)
exponentiation: use ^ rather than **, ², or ³.
Fibonacci (not fibonacci)
floor (not Floor, unless at the beginning of a sentence). Floor(x) is often denoted by [x].
Gamma (not GAMMA), for the Gamma function.
GCD (not g.c.d. or G.C.D.; not hcf or h.c.f. or HCF or H.C.F.): we prefer either GCD or gcd.
generalize (not generalise - the OEIS uses US spelling)
groups: The OEIS uses the modern names for groups, as in the ATLAS of Finite Groups and the International Tables for Crystallography. In particular, the dihedral group of order 2n is denoted by D_{2n}. (The old name was D_n.)
HCF, see GCD
i, not I, for , the imaginary unit. You can always add 'where i is the imaginary unit' to ensure nobody mistakes it for a summation index.
iff is a perfectly acceptable abbreviation for "if and only if"
Im, not im, for imaginary part of a complex number
independent (not independant)
infinity (not Inf, inf), but oo is allowed, especially in formulas
Integral_{x=1..2} f(x) dx or Integral_{x=0..oo} f(x) dx
The Iverson bracket is very useful in formulas: if P is some property or condition, [P] = 1 if P is true, [P] = 0 if P is false. So you can say "+ 6*[n == 0 (mod 3)] where [] is the Iverson bracket" as a shorthand for "except you have to add 6 if n is a multiple of 3".
J. S. Bach (not J.S. Bach - a period should be followed by a space, except in hyphenated names like J.-P. Serre)
l (el): try to avoid l in formulas, in many fonts it looks exactly like 1 (one) or I (capital i). Use k or m instead.
labeled, labeling (not labelled, labelling - the OEIS uses US spelling)
LCM (not l.c.m. or L.C.M.): we prefer either LCM or lcm.
lim_{n->P} for limit (use Limit_{n->P} at start of a sentence)
log_2(x) for logs to base 2
log(n) (not ln(x) or Log[x]) for logs to base e
log_10(x) for logs to base 10
< (not lrth)
<= (not \le or \leq)
multiplication sign: use * rather than X, ·, a period (dot), or ×. Both 6n^2+17n+1 and 6*n^2+17*n+1 are acceptable.
n X n (not n x n, not nXn, not n by n, not n-by-n)
n-th, m-th, i-th, j-th, etc. (not nth, mth, ith, jth)
neighbor (not neighbour - the OEIS uses US spelling)
nilpotent (not nil-potent)
noncomposite (not non-composite)
nondecreasing (not non-decreasing)
nonempty (not non-empty)
nonincreasing (not non-increasing)
nonnegative (not non-negative)
nonpositive (not non-positive)
nonprime (not non-prime)
nonsquare (not non-square)
nonsquarefree (not non-squarefree)
nontrivial (not non-trivial)
nonunit (not non-unit)
nonzero (not non-zero)
occurring (not occuring or ocurring)
octagon, not 8-gon: prefixes are used for figurate numbers below 9
p(n) should be prime(n)A000040, partition(n)A000041, or something unrelated
phi for the golden ratio (sometimes tau), phi(n) for the Euler totient function A000010
Pi for 3.141592653... (not pi)
prime(n) is the n-th prime A000040(n), and better than p(n)
prime(n) (preferred), or p(n), is usually the n-th prime, but both prime(n) and p(n) are also used in other ways
prime-indexed prime (not prime-index prime)
Product_{k=a..b} for product notation (always use capital "P"), e.g., Product_{k=1..n} a(k)
proved (not proven)
Re, not re, for the real part of a complex number
recurrence (not recurence or reccurrence)
recurring (not recuring or reccurring)
semiprime (not semi-prime)
sin(x) (not Sin[x])
spaces: follow the end of a sentence with one space or two, either way is acceptable
sqrt(x) (not Sqrt[x] or √x)
squarefree (not square-free)
submatrix (not sub-matrix)
Sum_{k=a..b} for summation notation (always use capital "S"), e.g., Sum_{n>0} a(n)
Sum_{i=1..n} Sum_{j=1..i} f(i,j,n) for a double sum
tetranacci (not Tetranacci)
triangular (not Triangular, unless at the beginning of a sentence)
tribonacci (not Tribonacci)
zeroth (not zeroeth)
zeros (not zeroes)
zeta (for lower case zeta, as in Riemann zeta function): use zeta not Zeta
Non-ASCII characters
Don't use non-ASCII characters! They just cause trouble, and the system may ignore a line containing a non-ASCII character without warning you.
For example, do not use Greek letters (π, Σ), unusual symbols like not-equal-to (≠), less-than-or-equal-to (≤), greater-than-or-equal-to (≥), the 3-dot ellipsis character (…), etc.
Exceptions: titles of published works and names of authors, OEIS contributors, places, institutions, etc. whose proper spelling requires non-ASCII characters; for example:
『Geneviève Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.』
A033308 Decimal expansion of Copeland-Erdős constant: concatenate primes. (Links section: Mikołaj Morzy, Tomasz Kajdanowicz, Przemysław Kazienko, ...)
Grammar
The possessive of a singular noun is formed by adding an apostrophe and an S, even if the noun ends in S. Write "Lucas's theorem", not "Lucas' theorem" (or perhaps "the Lucas theorem"). This rule is not universally accepted, so when directly quoting material for which the original editor did not abide by it, the quotation should not be altered.
Common mistakes in English
Using a(n) to refer to the whole sequence. That's wrong, a(n) is the n-th term.
To refer to the full sequence, use {a(n)} or the A-number, such as A123456.
"allows to" should be changed to "allows us to" or "allows one to"
"amount" (referring to a number) should be "number"
WRONG: "The amount of primes in ..."
CORRECT: "The number of primes in ..."
CORRECT: "The amount of butter in the cake ..."
WRONG: "The amount of eggs in the cake ..."
CORRECT: "The number of eggs in the cake ..."
"be integer" is wrong.
WRONG: "Let k be integer"
CORRECT: "Let k be an integer"
"couples" instead of "pairs"
WRONG: "a(n) is the number of couples of binary matrices..."
CORRECT: "a(n) is the number of pairs of binary matrices..."
"counts the number of" is wrong. Unless you are working in a hardware store, you don't count numbers.
WRONG: "a(n) counts the number of ..."
CORRECT: "a(n) is the number of ..."
CORRECT: "a(n) gives the number of ..."
"Except a(5) = 91, ..." is wrong
WRONG: "Except a(5) = 91, all terms are ..."
CORRECT: "Except for a(5) = 91, all terms are ..."
WRONG: "Except the number 2 ..."
CORRECT: "Except for the number 2 ..."
"fulfills" is almost always wrong: the correct English word is "satisfies". For some reason, this error is very common among French and German speakers.
"greater or equal" is wrong
WRONG: "greater or equal to n"
WRONG: "greater or equal n"
CORRECT: "greater than or equal to n"
CORRECT: ">= n"
"its" versus "it's":
"Its", no apostrophe, is the possessive of the pronoun "it": "the sequence is new, and its author is J. Smith"
"It's", with an apostrophe, is short for "it is" or "it has": "the sequence is new, but it's wrong"
"less or equal" is wrong
WRONG: "less or equal to n"
WRONG: "less or equal n"
CORRECT: "less than or equal to n"
CORRECT: "<= n"
"respectfully" when you should have said "respectively":
WRONG! "The terms congruent to 0, 1, 2 (mod 3) are respectfully given by A000001, A000002, A000003"
CORRECT: "The terms congruent to 0, 1, 2 (mod 3) are respectively given by A000001, A000002, A000003"
CORRECT: "The terms congruent to 0, 1, 2 (mod 3) are given by A000001, A000002, A000003, respectively"
"triplets" when you should have said "triples"
WRONG: "a(n) is the number of triplets of binary matrices..."
CORRECT: "a(n) is the number of triples of binary matrices..."
Similarly, "quadruple" is preferred over "quadruplet", and so on.
"unique" when you should have said "distinct"
WRONG: The list "Tom, Tom, Mary, John" contains three unique names"
CORRECT: The list "Tom, Tom, Mary, John" contains three distinct names"
Technical definitions
Divisors - often a source of confusion!
divisors: numbers d in the range 1 <= d <= n which divide n (A000005, A000203)
aliquot divisors, aliquot parts: numbers d < n which divide n (see A032741, A001065)
proper divisors, nontrivial divisors: officially these terms refer to divisors d of n with 1 < d < n (A070824), but are often used incorrectly for divisors d of n with 1 <= d < n (A032741, A001065). You should always specify which definition you are using.
prime divisors vs. divisors: a common mistake is to say "divisor" when you mean "prime divisor" or even "prime factor". (Prime factors may occur with multiplicity, while divisors may include p, p^2, ... but not the same p twice: prime divisors is a synonym for prime factors without repetition.)
omega(n) and Omega(n): omega(n) is the number of distinct primes dividing n (A001221), whereas Omega(n) or bigomega(n) counts them with multiplicity (A001222)
What does "mod n" mean? Many contributors to the OEIS find this confusing.
a == b (mod c) means that a-b is a multiple of c (one reads this as "a is congruent to b mod c"). So 12 == 8 (mod 2).
a mod c = b means that the remainder when a is divided by c is b. So 12 mod 2 is 0.
Sequences with conjectured terms
In principle, the terms shown in an OEIS entry should have been proved to be correct and complete as far as they are shown. For example, in the list of Mersenne primes, A000043, we don't include terms which are known to be in the sequence if it is possible that there are earlier terms which have not yet been found (although such terms are mentioned in the Comments or Extensions sections).
What do we do when there are terms in a sequence which are only conjectural?
The most common situation occurs when we know a certain number of terms, but we have only conjectures for the next few terms. In this case we give the terms that are certain in the Data section and the conjectured terms in the Comments or Extensions. This is the rule that we use to handle most cases. Example: A000952, numbers n == 2 (mod 4) for which a conference matrix of order n exists. It is only a conjecture that the next term is 66.
More than enough terms are known to fill three lines in the Data section, but there are gaps further along in the sequence. In this case we give the known terms (up to the first gap) in a b-file, and all the terms - with gaps, question marks, or ranges for the uncertain terms - in an a-file. Example: A072942, which has both a b-file and an a-file. A046057 is a sequence from group theory which has an a-file although we don't know enough terms for certain to give a b-file.
We only know a few terms for certain, but there is a conjectured generating function (which may or may not be correct). In this case we sometimes give two sequences, one for the known values and one for the sequence produced by the generating function. Example: A008368, arising from the face-centered cubic lattice, and A023054, from the proposed generating function.
An extreme example is the sequence of Riesel numbers, A076337, in which only the first term (509203) is known for certain. This violates all our rules, but is included in the OEIS because it is an important problem in number theory, and in the hope that having an entry for it in the OEIS will one day lead to the computation of further terms. The most likely extension is given by A101036. We hope that one day it will be possible to merge the two entries.
Two further extreme examples: If we were to insist on giving only terms which are known for certain, neither sequence would exist. Because of the importance of these problems, we have made exceptions and relaxed the rules.
The Brier numbers, A076335. Seven terms are known, but it is only a conjecture that they are the first seven terms. Even the first term shown is only conjectured to be the smallest.
The minimal number of polygonal pieces needed for dissecting a regular polygon with n sides into an equilateral triangle of the same area: A110000. This is a lovely problem, but only the trivial term a(3)=1 is known for certain. The other terms listed are just upper bounds. Again we hope that including this sequence in the OEIS will lead to the computation of further terms.
A000373 is a different kind of example. Here there is an explicit formula for a(n), and we can compute as many terms as we wish, but it is only a conjecture that this is the answer to a question of Yuri Manin about Moufang loops.
Probable primes. We take the point of view that that numbers which at present are known only to be probable primes will eventually be shown to be primes, so we don't regard sequences involving such numbers as conjectural. For example, see A004061, numbers n such that (5^n-1)/4 is prime. Whenever possible, you should nonetheless mention which terms only correspond to probable primes.
such as Goldbach's conjecture, the 3x+1 conjecture, etc., the OEIS is not the place to publish it. First publish your proof in a (reputable) mathematics journal.
, the imaginary unit. You can always add 'where i is the imaginary unit' to ensure nobody mistakes it for a summation index.