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A038538 Number of semisimple rings with n elements. 8
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; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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).

REFERENCES

T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(p^k) = A004101(k).

For all n, a(A002110(n)) = a(A005117(n)) = 1.

PROG

(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.

A004101(n) = v004101from1[n];

vecproduct(v) = { my(m=1); for(i=1, #v, m *= v[i]); m; };

A038538(n) = vecproduct(apply(e -> A004101(e), factorint(n)[, 2])); \\ Antti Karttunen, Nov 18 2017

CROSSREFS

Cf. A002110, A004101, A005117, A027623, A052305.

Sequence in context: A319786 A321271 A305193 * A293515 A326622 A292777

Adjacent sequences:  A038535 A038536 A038537 * A038539 A038540 A038541

KEYWORD

nonn,nice,mult

AUTHOR

Paolo Dominici (pl.dm(AT)libero.it)

STATUS

approved

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Last modified April 10 16:16 EDT 2021. Contains 342845 sequences. (Running on oeis4.)