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A384912 The number of unordered factorizations of n into exponentially squarefree prime powers (A384419).

%I A384912 #8 Jun 12 2025 10:24:38
%S A384912 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,
%T A384912 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,
%U A384912 1,1,1,6,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1
%N A384912 The number of unordered factorizations of n into exponentially squarefree prime powers (A384419).
%C A384912 First differs from A384913 at n = 64.
%H A384912 Amiram Eldar, <a href="/A384912/b384912.txt">Table of n, a(n) for n = 1..10000</a>
%F A384912 Multiplicative with a(p^e) = A073576(e).
%F A384912 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)).
%e A384912 a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with squarefree exponents 1 and 2.
%t A384912 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 *)
%t A384912 f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384912 (PARI) s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*issquarefree(d)) * s(n-j))/n);
%o A384912 a(n) = vecprod(apply(s, factor(n)[, 2]));
%Y A384912 Cf. A005117, A073576, A209061, A384419.
%Y A384912 Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384913, A384914, A384915, A384916.
%K A384912 nonn,easy,mult
%O A384912 1,4
%A A384912 _Amiram Eldar_, Jun 12 2025