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%I #33 Jul 13 2022 17:57:45
%S 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,
%T 2
%N Number of commutative rings with n elements.
%C These rings do not necessarily contain an identity element.
%C 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. Adams-Watters_, Jul 10 2012
%H Simon R. Blackburn and K. Robin McLean, <a href="https://arxiv.org/abs/2107.13215">The enumeration of finite rings</a>, 2022 preprint. arXiv:2107.13215 [math.CO]
%H A. V. Lelechenko, <a href="http://arxiv.org/abs/1305.1639">Parity of the number of primes in a given interval and algorithms of the sublinear summation</a>, arXiv preprint arXiv:1305.1639, 2013
%H C. Noebauer, <a href="https://web.archive.org/web/20080111141811/http://www.algebra.uni-linz.ac.at/~noebsi/">Home page</a> [Archived copy as of 2008 from web.archive.org]
%H Christof Noebauer, <a href="ftp://www.algebra.uni-linz.ac.at/pub/noebauer/smallrings.ps.gz">The numbers of small rings</a> (<a href="ftp://ftp.mathe2.uni-bayreuth.de/axel/papers/noebauer:the_number_of_small_rings.ps">PostScript</a>).
%H C. Noebauer, <a href="ftp://www.algebra.uni-linz.ac.at/pub/noebauer/thesis.ps.gz">Thesis on the enumeration of near-rings</a>
%H Bjorn Poonen, <a href="http://www-math.mit.edu/~poonen/papers/moduli.pdf">The moduli space of commutative algebras of finite rank</a>, J. Eur. Math. Soc. (JEMS) 10:3 (2008), pp. 817-836. <a href="http://arxiv.org/abs/math/0608491">arXiv:0608491</a> [This contains an error, see Blackburn & McLean]
%F a(p^n) = p^(2/27 * n^3 + O(n^2.5)), see Blackburn & McLean. - _Charles R Greathouse IV_, Jul 13 2022
%Y Cf. A027623, A037291.
%K nonn,nice,more,hard,mult
%O 1,2
%A _Christian G. Bower_, Jun 15 1998
%E a(16) from Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.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)
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