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

Offset: 1

Author

Amiram Eldar, Jun 12 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A384913 at n = 64.

Examples

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

Links

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

Crossrefs

Cf. A005117, A073576, A209061, A384419.

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

Programs

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]

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]));

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)).