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A027623
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a(0) = 1; for n > 0, a(n) = number of rings with n elements.
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23
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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
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OFFSET
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0,3
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COMMENTS
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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.
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
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LINKS
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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).
Eric Weisstein's World of Mathematics, Ring.
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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--------------------------------------------------------------------
| Rings with | with 1 | without 1 | with 1 or |
| n elements | | | without 1 |
--------------------------------------------------------------------
--------------------------------------------------------------------
--------------------------------------------------------------------
| noncommutative | | | A037234 with a(0) = 0 |
--------------------------------------------------------------------
(End)
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KEYWORD
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nonn,nice,hard,more,mult
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AUTHOR
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EXTENSIONS
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a(16) from Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000
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STATUS
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approved
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