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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.uni-linz.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 457-464, 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), 149-149.

C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64 (1980) p. 13, 1980, see esp. p. 21.

Yang-Hui 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 near-rings

Christof Noebauer, The Numbers of Small Rings (PostScript).

R. Raghavendran, Finite associative rings, Compositio Mathematica vol 21 no 2 (1969) p 195-229.

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.uni-linz.ac.at), Sep 29 2000

STATUS

approved

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Last modified October 19 10:51 EDT 2019. Contains 328216 sequences. (Running on oeis4.)