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 A037234 #27 Feb 16 2025 08:32:37
%S A037234 0,1,2,2,11,2,4,2,52,11,4,2,22,2,4,4,390,2,22,2,22,4,4,2,104,11,4,59,
%T A037234 22,2,8,2
%N A037234 a(n) = number of rings with n elements.
%C A037234 From _M. F. Hasler_, Jan 05 2021 (Start)
%C A037234 This entry complements the "main entry" A027623 which for all n >= 1 also gives the number of rings with n elements, but which has A027623(0) = 1 by explicit definition. (There is no ring with no elements, since a ring is an abelian group and therefore must have at least the 0 element.)
%C A037234 a(32) is presently unknown: see A027623 for lower bounds and values a(n) for n > 32. (End)
%H A037234 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), doi:10.1007/BF00968650.
%H A037234 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Ring.html">Ring</a>.
%F A037234 From _M. F. Hasler_, Jan 05 2021: (Start)
%F A037234 a(p) = 2 for any prime p.
%F A037234 a(m n) = a(m) a(n) when gcd(m,n) = 1. (Multiplicativity.)
%F A037234 a(p^2) = 11 for any prime p.
%F A037234 a(p^3) = 3p + 50 for any odd prime p [Antipkin & Elizarov]. (End)
%e A037234 From _Bernard Schott_, Jan 06 2021: (Start)
%e A037234 a(1) = 1: The ring with only one element, 0, is called the zero ring.
%e A037234 a(2) = 2: These two rings of order 2 with elements {0, a} form an abelian group for operator +: 0+0 = 0, 0+a = a+0 = a, a+a = 0.
%e A037234 - The first ring is obtained for multiplication defined by: 0*0 = 0*a = a*0 = 0, a*a = a. This ring is isomorphic to the field Z/2Z.
%e A037234 - The second ring is given for 0*0 = 0*a = a*0 = a*a = 0. Here a is a divisor of 0. (End)
%o A037234 (PARI) apply( A037234(n, e=0)=if( !e, vecprod([call(self(),f) | f <- factor(n)~]), e<3, [if(n,2), 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 A037234 A027623 is the main entry for this sequence.
%K A037234 nonn,mult,hard,more
%O A037234 0,3
%A A037234 _N. J. A. Sloane_