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A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

%I A384914 #7 Jun 12 2025 10:24:45
%S A384914 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,
%T A384914 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,
%U A384914 1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1
%N A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).
%C A384914 First differs from A203640, A295658 and A365333 at n = 64, from A043289 and A053164 at n = 81, and from A063775 at n = 512.
%H A384914 Amiram Eldar, <a href="/A384914/b384914.txt">Table of n, a(n) for n = 1..10000</a>
%F A384914 Multiplicative with a(p^e) = A001156(e).
%F A384914 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.08451356983124311685..., where f(x) = (1-x) / Product_{k>=1} (1-x^(k^2)).
%e A384914 a(16) = 2 since 4 has 2 factorizations: 2^1 * 2^1 * 2^1 * 2^1 and 2^4, with exponents 1 and 4 that are squares.
%t A384914 s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * Boole[IntegerQ[Sqrt[d]]], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
%t A384914 f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384914 (PARI) s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*issquare(d)) * s(n-j))/n);
%o A384914 a(n) = vecprod(apply(s, factor(n)[, 2]));
%Y A384914 Cf. A000290, A001156, A197680, A323520.
%Y A384914 Cf. A043289, A053164, A063775, A203640, A295658, A365333.
%Y A384914 Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384915, A384916.
%K A384914 nonn,easy,mult
%O A384914 1,16
%A A384914 _Amiram Eldar_, Jun 12 2025