A037289 Number of commutative rings with n elements.

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

Offset: 1

Author

Christian G. Bower, Jun 15 1998

Keywords

• nonn
• nice
• more
• hard
• mult

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. Adams-Watters, Jul 10 2012

Links

• 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 near-rings
• Bjorn Poonen, The moduli space of commutative algebras of finite rank, J. Eur. Math. Soc. (JEMS) 10:3 (2008), pp. 817-836. arXiv:0608491 [This contains an error, see Blackburn & McLean]

Crossrefs

Cf. A027623, A037291.

Formula

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

Extensions

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)