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Search: id:A027623
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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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+0
10
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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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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 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]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 20 2008
"This completes the discussion and we see that there are in all 3+2+2+3+1=11 mutually nonisomorphic rings of order p^2" [Raghavendran, p. 228] - R. J. Mathar, Apr 17 2008
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REFERENCES
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R. Ballieu [ Math. Rev. 9, 267; see also Math. Rev. 51#5655 ] showed a(8)=52, a(p^3)=3p+50 if p is odd prime.
C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64, p. 13, 1980, see esp. p. 21.
V. G. Antipkin and V. P. Elizarov [Math. Rev. 84d:16025]
R. Raghavendran, Finite associative rings, Compositio Mathematica, vol 21, no 2 (1969) p195-229.
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LINKS
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C. Noebauer, Home page
C. Noebauer, The Numbers of Small Rings
C. Noebauer, Thesis on the enumeration of near-rings
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Christof Noebauer, The Numbers of Small Rings (PostScript).
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. Raghavendran, Finite associative rings, Compositio Mathematica vol 21 no 2 (1969) p 195-229.
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EXAMPLE
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The 11 rings of order 4 (from Christian G. Bower bowerc(AT)usa.net): 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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CROSSREFS
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Cf. A037289, A037291.
Sequence in context: A000371 A081088 A001038 this_sequence A037234 A090525 A126806
Adjacent sequences: A027620 A027621 A027622 this_sequence A027624 A027625 A027626
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KEYWORD
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nonn,nice,hard,mult
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AUTHOR
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njas, R. K. Guy
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EXTENSIONS
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More terms from Christian G. Bower (bowerc(AT)usa.net), Jun 15 1998. a(16) from Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29, 2000
Christof Noebauer also reports that 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)
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