On-Line Encyclopedia of Integer Sequences

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The On-Line Encyclopedia of Integer Sequences ( OEIS ; German Online Encyclopedia of Number Sequences ) is an English-language database of sequences of whole numbers ( integer sequences ) that can be searched over the Internet. It is an often used tool and an important source in mathematical research.



The encyclopedia is a database that collects information about sequences of integers of interest in mathematics. In mid-March 2018, the database contained over 300,000 sequences of numbers. Each entry contains the first parts of the sequence and a motivation for this sequence, keywords, references, hyperlinks, programs for Mathematica , Maple , PARI and other instructions for creating the sequence, references to related sequences, and much more.

The database can be searched for both keywords and substrings.

Internal format

The sequences are described in the database in a pure ASCII line format. Each line begins with a percent sign , a letter code for the type of partial information, the number of the sequence and the respective partial information. For example, sequence A004002 is stored as follows:

%I M3010
%S 1,3,15,3814279
%N Benford numbers: a(n)=e^e^...^e (n times) rounded to nearest integer.
%C The next term, a(4) ~ 2.3315*10^1656520, has 1656521 decimal digits and is therefore too large to be included. [Rephrased by _M. F. Hasler_, May 01 2013]
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D P. R. Turner, Will the "real" real arithmetic please stand up?, Notices Amer. Math. Soc., 38 (1991), 298–304.
%F a(n) = round(e^e^...^e), where e occurs n times, a(0) = 1 (= e^0). - _Melissa O'Neill_, Jul 04 2015
%t Round[NestList[Power[E, #] &, 1, 3]] (* _Melissa O'Neill_, Jul 04 2015 *)
%Y Cf. A056072, A225053.
%Y Cf. A073236. - _Melissa O'Neill_, Jul 04 2015
%K nonn
%O 0,2
%A _N. J. A. Sloane_


The On-Line Encyclopedia of Integer Sequences is considered by many to be the primary reference in the field of sequences of integers. Most of the papers in which such sequences of numbers appear contain references to the database.

The database is the largest of its kind in the world and receives several thousand inquiries every day. Their success is due in part to the fact that the data can be accessed free of charge.

Because of the OEIS, some mathematical discoveries were made. For example, someone (RD North from Colorado Springs) discovered that the number formed when the Leibniz series was cut off for Pi (partial sum of the Leibniz series) had shorter and longer sections of decimal places that matched those of Pi. Jonathan Borwein examined the decimal places of the difference and found that when divided by 2 it matched the sequence of Euler's numbers from OEIS. Together with colleagues, he was able to use this to give an exact formula for the error term of the approximation of Pi by the truncated Leibniz series.


Neil Sloane began collecting integer sequences in 1964 to make his work in combinatorics easier (presumably based on similar sequences from his dissertation on neural networks). He published parts of the database in book form twice:

  1. A Handbook of Integer Sequences (1973, ISBN 0-12-648550-X ), with 2372 episodes.
  2. The Encyclopedia of Integer Sequences (1995, with the mathematician Simon Plouffe ), ISBN 0-12-558630-2 ), with 5488 sequences.

These books were enthusiastically received, and after the second publication the collection grew too large (doubling in size one year after the second publication) to be republished as a book, and when the database contained 16,000 entries, Sloane decided that Making data accessible online, first as an email service (1995) and soon thereafter as a web service (1996). Since then, the database has grown by around 10,000 entries per year. Sloane itself generated the entries for over 170,000 episodes (2015). In 2015 around 4,000 users were registered with OEIS.

After Neil Sloane had managed his database for nearly 40 years, a group of editors did much of the maintenance in 2002. Since 2009 it has been in the form of a wiki with around 100 volunteer editors. The ultimate authority for accepting or rejecting an entry remains Neil Sloane, and since early 2006 the frequent acceptance of new episodes has turned into a relatively restrictive policy.

As an offshoot of his database work, Sloane founded the Journal of Integer Sequences in 1998 . In October 2009, the intellectual property and the operation of the servers went to the OEIS Foundation, which was established for this purpose .

Sloane's Gap

Sloane's Gap

If you show in a diagram the number of different sequences listed in the database in which a natural number n appears, the point cloud for this frequency N n approximately follows the curve N n = 253,000,000 / n 1.33 . A curiosity in this cloud / curve represents a gap (engl. Gap ) in this represents, which can be observed in particular for the numbers from 300 to 10,000. This gap apparently divides the mathematically interesting numbers that exist in very many sequences from the uninteresting ones. For example, almost all (99.7%) of the between 300 and 10,000 prime numbers are in the upper part of the curve. Around 95% of all square numbers between 300 and 10,000 can also be found there. While the curve itself corresponds to the expected value, the gap cannot yet be explained in a purely mathematical way. It is therefore possibly also due to the popularity of certain sequences of numbers in mathematical research and thus to social factors.

Individual evidence

  1. Jonathan Borwein, Peter Borwein , K. Dilcher: Pi, Euler Numbers and Asymptotic Expansions, American Mathematical Monthly, Volume 96, 1989, pp. 681-687
  2. Interview with Quanta Magazine 2015
  3. Transfer of IP in OEIS to The OEIS Foundation Inc. ( Memento of the original from December 6, 2013 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / oeisf.org
  4. Nicolas Gauvrit, Jean-Paul Delahaye, Hector Zenil: Sloane's Gap: Thu Mathematical and Social Factors Explain the distribution of Numbers in the OEIS? Retrieved February 25, 2014.

Web links