A384816 Sum of the cubes of the indices of distinct prime factors of n.

0, 1, 8, 1, 27, 9, 64, 1, 8, 28, 125, 9, 216, 65, 35, 1, 343, 9, 512, 28, 72, 126, 729, 9, 27, 217, 8, 65, 1000, 36, 1331, 1, 133, 344, 91, 9, 1728, 513, 224, 28, 2197, 73, 2744, 126, 35, 730, 3375, 9, 64, 28, 351, 217, 4096, 9, 152, 65, 520, 1001, 4913, 36, 5832, 1332, 72, 1, 243, 134, 6859, 344, 737, 92

Offset

1,3

Links

Table of n, a(n) for n=1..70.

Index entries for sequences computed from indices in prime factorization.

Formula

If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^3), where pi = A000720.

G.f.: Sum_{k>=1} k^3 * x^prime(k) / (1 - x^prime(k)).

Mathematica

Table[Plus @@ (PrimePi[#[[1]]]^3 & /@ FactorInteger[n]), {n, 70}]
nmax = 70; CoefficientList[Series[Sum[k^3 x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Prog

(PARI) a(n) = my(f=factor(n)[, 1]); sum(k=1, #f~, primepi(f[k])^3); \\ Michel Marcus, Jun 10 2025

Crossrefs

Cf. A000720, A005064, A066328, A332385.

Sequence in context: A050458 A358877 A125166 * A211785 A075151 A075155

Adjacent sequences: A384813 A384814 A384815 * A384817 A384818 A384819

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 10 2025

Status

approved