cp's OEIS Frontend

A384785 The number of unordered factorizations of the n-th cubefull number into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

%I A384785 #10 Jun 10 2025 11:34:38
%S A384785 1,1,1,1,1,2,1,1,2,1,1,3,1,1,4,1,1,2,1,1,5,1,1,2,1,1,6,2,1,1,1,1,1,1,
%T A384785 2,1,1,9,1,1,2,1,2,3,1,3,1,2,10,1,1,1,2,1,1,2,1,4,1,2,2,2,13,1,1,2,1,
%U A384785 1,4,1,3,1,2,2,1,1,1,5,1,1,1,1,2,3,17,2
%N A384785 The number of unordered factorizations of the n-th cubefull number into 1 and prime powers p^e where p is prime and e >= 3 (A246549).
%C A384785 The positive values of the multiplicative function f(n) with f(p^e) = A008483(e). Or, equivalently, a(n) is the value of this function at A036966(n).
%H A384785 Amiram Eldar, <a href="/A384785/b384785.txt">Table of n, a(n) for n = 1..10000</a>
%e A384785 a(6) = 2 since the 6th cubefull number, A036966(6) = 64, has 2 factorizations: 2^3 * 2^3 and 2^6.
%e A384785 a(12) = 3 since the 12th cubefull number, A036966(12) = 256, has 3 factorizations: 2^3 * 2^5, 2^4 * 2^4, and 2^8.
%t A384785 f[p_, e_] := PartitionsP[e] - PartitionsP[e-1] - PartitionsP[e-2] + PartitionsP[e-3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq[lim_] := Module[{cub = Union[Flatten[Table[i^3*j^4*k^5, {k, 1, Surd[lim, 5]}, {j, 1, Surd[lim/k^5, 4]}, {i, 1, Surd[lim/(j^4*k^5), 3]}]]]}, Select[s /@ cub, # > 0 &]]; seq[10^5]
%o A384785 (PARI) s(n) = vecprod(apply(x -> numbpart(x)-numbpart(x-1)-numbpart(x-2)+numbpart(x-3), factor(n)[, 2]));
%o A384785 cubs(lim) = {my(c = List()); for(k = 1, sqrtnint(lim, 5), for(j = 1, sqrtnint(lim \ k^5, 4), for(i = 1, sqrtnint(lim \ (j^4*k^5), 3), listput(c, i^3*j^4*k^5)))); Set(c); }
%o A384785 list(lim) = {my(c = cubs(lim), v = List(), s1); for(k = 1, #c, s1 = s(c[k]); if(s1 > 0, listput(v, s1))); Vec(v);}
%Y A384785 Cf. A008483, A036966, A246549, A384783 (powerful analog), A384786.
%K A384785 nonn,easy
%O A384785 1,6
%A A384785 _Amiram Eldar_, Jun 10 2025