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Search: id:A038538
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| A038538 |
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Number of semisimple rings with n elements. |
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+0
5
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| 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 13, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 6, 6, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Enumeration uses Wedderburn-Artin theorem and fact that a finite division ring is a field.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375 =3*5^3 both have prime signature (3,1).
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REFERENCES
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T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag.
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FORMULA
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Dirichlet generating function: sum( a(n) / n^s, n=1..infinity) = product ( 1/(1-p^(-r*m^2*s)), r= 1..infinity, m=1..infinity, p prime ) = product ( zeta(k*s)^A046951(n), k=1..infinity).
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PROGRAM
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(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X-X^2))[n] (from R. Stephan)
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CROSSREFS
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Cf. A027623, A046951, A052305. a(p^k)=A004101. a(A002110)=1.
Sequence in context: A008479 A107345 A000688 this_sequence A088529 A136565 A086291
Adjacent sequences: A038535 A038536 A038537 this_sequence A038539 A038540 A038541
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KEYWORD
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nonn,nice,mult
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AUTHOR
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Paolo Dominici (pl.dm(AT)libero.it)
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