A370328 The number of powerful divisors of the powerful numbers.

1, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 6, 4, 4, 6, 2, 3, 7, 8, 2, 4, 6, 9, 4, 5, 8, 10, 2, 8, 3, 2, 6, 8, 12, 4, 4, 6, 9, 2, 12, 4, 12, 6, 4, 6, 8, 10, 2, 15, 8, 2, 6, 10, 9, 10, 4, 6, 14, 4, 4, 16, 6, 3, 6, 2, 6, 4, 4, 10, 12, 2, 18, 8, 12, 2, 8, 15, 12, 8, 11, 4, 7

Offset: 1

Author

Amiram Eldar, Feb 15 2024

Keywords

• nonn
• easy

Comments

The product of the exponents of the prime factorization of the powerful numbers.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001694, A005361, A306458, A363194, A370329.

Programs

Mathematica

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[n == 1 || Min[e] > 1, Times @@ e, Nothing]]; Array[f, 2500]

PARI

lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1 || vecmin(e) > 1, print1(vecprod(e), ", ")));}

Formula

a(n) = A005361(A001694(n)).

a(n) = A000005(A306458(n)).