A037234 a(n) = number of rings with n elements.

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, 22, 2, 8, 2

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• hard
• more

Comments

From M. F. Hasler, Jan 05 2021 (Start)

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.)

a(32) is presently unknown: see A027623 for lower bounds and values a(n) for n > 32. (End)

Examples

			From _Bernard Schott_, Jan 06 2021: (Start)
a(1) = 1: The ring with only one element, 0, is called the zero ring.
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.
   - 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.
   - The second ring is given for 0*0 = 0*a = a*0 = a*a = 0. Here a is a divisor of 0. (End)
		

Links

• V. G. Antipkin and V. P. Elizarov, Rings of order p^3, Sib. Math. J. vol 23 no 4 (1982) pp 457-464, MR0668331 (84d:16025), doi:10.1007/BF00968650.
• Eric Weisstein's World of Mathematics, Ring.

Crossrefs

A027623 is the main entry for this sequence.

Programs

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

Formula

From M. F. Hasler, Jan 05 2021: (Start)

a(p) = 2 for any prime p.

a(m n) = a(m) a(n) when gcd(m,n) = 1. (Multiplicativity.)

a(p^2) = 11 for any prime p.

a(p^3) = 3p + 50 for any odd prime p [Antipkin & Elizarov]. (End)