

A037289


Number of commutative rings with n elements.


15



1, 2, 2, 9, 2, 4, 2, 34, 9, 4, 2, 18, 2, 4, 4, 162, 2, 18, 2, 18, 4, 4, 2, 68, 9, 4, 36, 18, 2, 8, 2
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OFFSET

1,2


COMMENTS

These rings do not necessarily contain an identity element.
This sequence is multiplicative. See the reference "The Numbers of Small Rings" below, which proves the result for all rings; restricting to commutative rings only makes the proof easier.  Conjecture by Mitch Harris, Apr 19 2005, proof found by Franklin T. AdamsWatters, Jul 10 2012


LINKS

Table of n, a(n) for n=1..31.
Simon R. Blackburn and K. Robin McLean, The enumeration of finite rings, 2022 preprint. arXiv:2107.13215 [math.CO]
A. V. Lelechenko, Parity of the number of primes in a given interval and algorithms of the sublinear summation, arXiv preprint arXiv:1305.1639, 2013
C. Noebauer, Home page [Archived copy as of 2008 from web.archive.org]
Christof Noebauer, The numbers of small rings (PostScript).
C. Noebauer, Thesis on the enumeration of nearrings
Bjorn Poonen, The moduli space of commutative algebras of finite rank, J. Eur. Math. Soc. (JEMS) 10:3 (2008), pp. 817836. arXiv:0608491 [This contains an error, see Blackburn & McLean]


FORMULA

a(p^n) = p^(2/27 * n^3 + O(n^2.5)), see Blackburn & McLean.  Charles R Greathouse IV, Jul 13 2022


CROSSREFS

Cf. A027623, A037291.
Sequence in context: A082838 A074961 A300450 * A037290 A155936 A167594
Adjacent sequences: A037286 A037287 A037288 * A037290 A037291 A037292


KEYWORD

nonn,nice,more,hard,mult


AUTHOR

Christian G. Bower, Jun 15 1998


EXTENSIONS

a(16) from Christof Noebauer (christof.noebauer(AT)algebra.unilinz.ac.at), Sep 29 2000, who reports that the sequence continues a(32) = ? (> 876), a(33) = 4, 4, 4, 81, 2, 4, 4, 68, 2, 8, 2, 18, 18, 4, 2, 324, 9, 18, 4, 18, 2, 72, 4, 68, 4, 4, 2, 36, 2, 4, 18 = a(63), a(64) = ? (> 12696)


STATUS

approved



