

A027623


a(0) = 1; for n > 0, a(n) = number of rings with n elements.


14



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, 22, 2, 8, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

Here a ring means (R,+,*): (R,+) is 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.
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.unilinz.ac.at), Sep 29 2000
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
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


LINKS

Table of n, a(n) for n=0..31.
V. G. Antipkin and V. P. Elizarov, Rings of order p^3, Sib. Math. J. vol 23 no 4 (1982) pp 457464, MR0668331 (84d:16025).
R. Ballieu [ Math. Rev. 0022841; see also Math. Rev. 51#5655] showed a(8)=52, a(p^3)=3p+50 if p is odd prime.
Grigore Călugăreanu, Rings with very few nilpotents, An. Sţiinţ. Univ. Al. I. Cuza Iaşi. Mat. (2018), 149149.
C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64 (1980) p. 13, 1980, see esp. p. 21.
YangHui He, Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.
A. V. Lelechenko, Parity of the number of primes in a given interval and algorithms of the sublinear summation, arXiv preprint arXiv:1305.1639 [math.NT], 2013.
C. Noebauer, The Numbers of Small Rings
C. Noebauer, Thesis on the enumeration of nearrings
Christof Noebauer, The Numbers of Small Rings (PostScript).
R. Raghavendran, Finite associative rings, Compositio Mathematica vol 21 no 2 (1969) p 195229.
Eric Weisstein's World of Mathematics, Ring


EXAMPLE

The 11 rings of order 4 (from Christian G. Bower): over C4: 1*1 = 0, 1 or 2; over C2 X C2 = <1> X <2>: (1*1,1*2,2*1,2*2) = 0000, 0001, 0002, 0012, 0102, 0112, 1002 or 1223.


CROSSREFS

Cf. A037289, A037291.
Sequence in context: A236369 A001038 A283454 * A037234 A141651 A213990
Adjacent sequences: A027620 A027621 A027622 * A027624 A027625 A027626


KEYWORD

nonn,nice,hard,more,mult


AUTHOR

N. J. A. Sloane, R. K. Guy


EXTENSIONS

More terms from Christian G. Bower, Jun 15 1998
a(16) from Christof Noebauer (christof.noebauer(AT)algebra.unilinz.ac.at), Sep 29 2000


STATUS

approved



