A384912 - OEIS

A384912

The number of unordered factorizations of n into exponentially squarefree prime powers (A384419).

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 9, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

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OFFSET

1,4

COMMENTS

First differs from A384913 at n = 64.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

Multiplicative with a(p^e) = A073576(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 2.1069024289184419840496..., where f(x) = (1-x) / Product_{k>=1} (1-x^A005117(k)).

EXAMPLE

a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with squarefree exponents 1 and 2.

MATHEMATICA

s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * Abs[MoebiusMu[d]], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n]; (* Jean-François Alcover at A073576 *)

f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PROG

(PARI) s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*issquarefree(d)) * s(n-j))/n);

a(n) = vecprod(apply(s, factor(n)[, 2]));

CROSSREFS

Cf. A005117, A073576, A209061, A384419.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384913, A384914, A384915, A384916.

Sequence in context: A322885 A292582 A384913 * A008479 A331178 A227350

Adjacent sequences: A384909 A384910 A384911 * A384913 A384914 A384915

KEYWORD

nonn,easy,mult

AUTHOR

Amiram Eldar, Jun 12 2025

STATUS

approved