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A038538
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Number of semisimple rings with n elements.
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11
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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
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OFFSET
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1,4
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COMMENTS
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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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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, pp. 274-276.
John Knopfmacher, Abstract analytic number theory, North-Holland, 1975, pp. 63-64.
T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, 2001.
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LINKS
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FORMULA
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Multiplicative with a(p^k) = A004101(k).
Dirichlet g.f.: Product_{k,m>=1} zeta(k*m^2*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.499616... = A244285 (see A123030 for a more precise asymptotic formula). (End)
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MATHEMATICA
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With[{emax = 7}, f[e_] := f[e] = Coefficient[Series[Product[1/(1 - x^(j*k^2)), {k, 1, Floor[Sqrt[emax]] + 1}, {j, 1, Floor[emax/k^2] + 1}], {x, 0, emax}], x, e]; a[1] = 1; a[n_] := Times @@ f /@ FactorInteger[n][[;; , 2]]; Array[a, 2^emax]] (* Amiram Eldar, Jan 31 2024, using code by Vaclav Kotesovec at A004101 *)
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PROG
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(PARI)
v004101from1 = [1, 2, 3, 6, 8, 13, 18, 29, 40, 58, 79, 115, 154, 213, 284, 391, 514, 690, 900, 1197]; \\ From the data-section of A004101.
vecproduct(v) = { my(m=1); for(i=1, #v, m *= v[i]); m; };
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CROSSREFS
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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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STATUS
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approved
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