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A384783 The number of unordered factorizations of the n-th powerful number into 1 and prime powers p^e where p is prime and e >= 2 (A025475).

%I A384783 #9 Jun 10 2025 11:34:23
%S A384783 1,1,1,1,2,1,1,2,1,1,4,1,2,1,1,1,1,4,2,1,1,1,1,1,2,7,2,1,2,1,1,1,2,2,
%T A384783 1,1,1,8,1,4,2,2,1,1,4,2,2,1,2,1,1,1,2,1,12,1,1,4,1,1,4,1,1,1,1,1,1,1,
%U A384783 2,4,1,4,1,1,1,2,2,2,2,14,1,4,1,1,7,1,2
%N A384783 The number of unordered factorizations of the n-th powerful number into 1 and prime powers p^e where p is prime and e >= 2 (A025475).
%C A384783 The positive terms in A188585.
%H A384783 Amiram Eldar, <a href="/A384783/b384783.txt">Table of n, a(n) for n = 1..10000</a>
%F A384783 a(n) = A188585(A001694(n)).
%e A384783 a(5) = 2 since the 5th powerful number, A001694(5) = 16, has 2 factorizations: 2^2 * 2^2 and 2^4.
%e A384783 a(11) = 4 since the 11th powerful number, A001694(11) = 64, has 4 factorizations: 2^2 * 2^2 * 2^2, 2^2 * 2^4, 2^3 * 2^3, and 2^6.
%t A384783 f[p_, e_] := PartitionsP[e] - PartitionsP[e-1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq[lim_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}]]]}, Select[s /@ pow, # > 0 &]]; seq[10^4]
%o A384783 (PARI) s(n) = vecprod(apply(x -> numbpart(x)-numbpart(x-1), factor(n)[, 2]));
%o A384783 pows(lim) = {my(p = List()); for(j = 1, sqrtnint(lim, 3), for(i = 1, sqrtint(lim \ j^3), listput(p, i^2 * j^3))); Set(p); }
%o A384783 list(lim) = {my(p = pows(lim), v = List(), s1); for(k = 1, #p, s1 = s(p[k]); if(s1 > 0, listput(v, s1))); Vec(v);}
%Y A384783 Cf. A001694, A025475, A188585, A384784, A384785 (cubefull analog).
%K A384783 nonn,easy
%O A384783 1,5
%A A384783 _Amiram Eldar_, Jun 10 2025