This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027623 #73 Feb 16 2025 08:32:35
%S A027623 1,1,2,2,11,2,4,2,52,11,4,2,22,2,4,4,390,2,22,2,22,4,4,2,104,11,4,59,
%T A027623 22,2,8,2
%N A027623 a(0) = 1; for n > 0, a(n) = number of rings with n elements.
%C A027623 Here a ring means (R,+,*): (R,+) is an abelian group, * is associative, a*(b+c) = a*b + a*c, (a+b)*c = a*c + b*c. Need not contain "1", * need not be commutative.
%C A027623 The sequence continues a(32) = ? (>18590), a(33) = 4, 4, 4, 121, 2, 4, 4, 104, 2, 8, 2, 22, 22, 4, 2, 780, 11, 22, 4, 22, 2, 118, 4, 104, 4, 4, 2, 44, 2, 4, 22 = a(63), a(64) = ? (> 829826). - Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000
%C A027623 The paper by Antipkin/Elizarov also gives the number a(p^3) of rings of order p^3. - Hans H. Storrer (storrer(AT)math.unizh.ch), Sep 16 2003
%C A027623 If n is a squared prime, there are 11 mutually nonisomorphic rings of order n [see Raghavendran, p. 228]. - _R. J. Mathar_, Apr 20 2008
%H A027623 V. G. Antipkin and V. P. Elizarov, <a href="http://dx.doi.org/10.1007/BF00968650">Rings of order p^3</a>, Sib. Math. J. vol 23 no 4 (1982) pp 457-464, MR0668331 (84d:16025).
%H A027623 R. Ballieu [ <a href="http://www.ams.org/mathscinet-getitem?mr=22841">Math. Rev. 0022841</a>; see also <a href="http://www.ams.org/mathscinet-getitem?mr=369422">Math. Rev. 51#5655</a>] showed a(8) = 52, a(p^3) = 3p + 50 if p is an odd prime.
%H A027623 Grigore Călugăreanu, <a href="http://math.ubbcluj.ro/~calu/3ni.pdf">Rings with very few nilpotents</a>, An. Sţiinţ. Univ. Al. I. Cuza Iaşi. Mat. (2018), p. 149.
%H A027623 C. R. Fletcher, <a href="http://www.jstor.org/stable/3615885">Rings of small order</a>, Math. Gaz. vol. 64 (1980) p. 13, 1980, see esp. p. 21.
%H A027623 Yang-Hui He and Minhyong Kim, <a href="https://arxiv.org/abs/1905.02263">Learning Algebraic Structures: Preliminary Investigations</a>, arXiv:1905.02263 [cs.LG], 2019.
%H A027623 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 [math.NT], 2013.
%H A027623 Desmond MacHale, <a href="https://doi.org/10.1080/00029890.2020.1820790">Are There More Finite Rings than Finite Groups</a>, Amer. Math. Monthly (2020) Vol. 127, Issue 10, 936-938.
%H A027623 C. Noebauer, <a href="ftp://www.algebra.uni-linz.ac.at/pub/noebauer/smallrings.ps.gz">The Numbers of Small Rings</a>.
%H A027623 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 A027623 Christof Noebauer, <a href="ftp://ftp.mathe2.uni-bayreuth.de/axel/papers/noebauer:the_number_of_small_rings.ps">The Numbers of Small Rings</a> (PostScript).
%H A027623 R. Raghavendran, <a href="http://www.numdam.org/item?id=CM_1969__21_2_195_0">Finite associative rings</a>, Compositio Mathematica vol 21 no 2 (1969) pp. 195-229.
%H A027623 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Ring.html">Ring</a>.
%H A027623 <a href="/index/Res#rings">Index to sequences related to rings</a>.
%e A027623 The 11 rings of order 4 (from _Christian G. Bower_):
%e A027623 over C4: 1*1 = 0, 1 or 2;
%e A027623 over C2 X C2 = <1> X <2>: (1*1,1*2,2*1,2*2) = 0000, 0001, 0002, 0012, 0102, 0112, 1002 or 1223.
%o A027623 (PARI) apply( A027623(n, e=0)=if( !e, vecprod([call(self(), f) | f <- factor(n)~]), e<3, [2^(n>0), 11][e], e==3, if(n>2, 3*sqrtnint(n, 3), 2)+50, n>2 || e>4, /*error*/("not yet implemented"), 390), [0..63]) \\ _M. F. Hasler_, Jan 05 2021
%Y A027623 From _Bernard Schott_, Mar 28 2021: (Start)
%Y A027623 --------------------------------------------------------------------
%Y A027623 | Rings with | with 1 | without 1 | with 1 or |
%Y A027623 | n elements | | | without 1 |
%Y A027623 --------------------------------------------------------------------
%Y A027623 | Commutative | A127707 | A342375 | A037289 |
%Y A027623 --------------------------------------------------------------------
%Y A027623 | Noncommutative | A127708 | A342376 | A209401 |
%Y A027623 --------------------------------------------------------------------
%Y A027623 | Commutative or | A037291 | A342377 | this sequence: a(0) = 1 |
%Y A027623 | noncommutative | | | A037234 with a(0) = 0 |
%Y A027623 --------------------------------------------------------------------
%Y A027623 (End)
%K A027623 nonn,nice,hard,more,mult
%O A027623 0,3
%A A027623 _N. J. A. Sloane_, _R. K. Guy_
%E A027623 More terms from _Christian G. Bower_, Jun 15 1998
%E A027623 a(16) from Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000