Search: id:A027623
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%I A027623
%S A027623 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
%N A027623 a(0) = 1; for n > 0, a(n) = number of rings with n elements.
%C A027623 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.
%C A027623 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
%C A027623 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
%C A027623 "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
%D A027623 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.
%D A027623 C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64, p. 13, 1980, 
               see esp. p. 21.
%D A027623 V. G. Antipkin and V. P. Elizarov [Math. Rev. 84d:16025]
%D A027623 R. Raghavendran, Finite associative rings, Compositio Mathematica, vol 
               21, no 2 (1969) p195-229.
%H A027623 C. Noebauer, <a href="http://www.algebra.uni-linz.ac.at/~noebsi/">Home 
               page</a>
%H A027623 C. Noebauer, <a href="ftp://www.algebra.uni-linz.ac.at/pub/noebauer/smallrings.ps.gz">
               The Numbers of Small Rings</a>
%H A027623 C. Noebauer, <a href="ftp://www.algebra.uni-linz.ac.at/pub/noebauer/thesis.ps.gz">
               Thesis on the enumeration of near-rings</a>
%H A027623 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Ring.html">Link to a section of The World of Mathematics.</a>
%H A027623 Christof Noebauer, <a href="ftp://ftp.mathe2.uni-bayreuth.de/axel/papers/
               noebauer:the_number_of_small_rings.ps">The Numbers of Small Rings</
               a> (PostScript).
%H A027623 V. G. Antipkin and V. P. Elizarov, <a href="http://dx.doi.org/10.1007/
               BF00968650">Rings of order p^3</a>, Sib. Math. J. vol 23 no 4 (1982) 
               pp 457-464, MR0668331 (84d:16025)
%H A027623 R. Raghavendran, <a href="http://www.numdam.org/item?id=CM_1969__21_2_195_0">
               Finite associative rings</a>, Compositio Mathematica vol 21 no 2 
               (1969) p 195-229.
%e A027623 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.
%Y A027623 Cf. A037289, A037291.
%Y A027623 Sequence in context: A000371 A081088 A001038 this_sequence A037234 A141651 
               A090525
%Y A027623 Adjacent sequences: A027620 A027621 A027622 this_sequence A027624 A027625 
               A027626
%K A027623 nonn,nice,hard,mult
%O A027623 0,3
%A A027623 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A027623 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
%E A027623 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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