A368040 The powerful part of the nonsquarefree numbers.

4, 8, 9, 4, 16, 9, 4, 8, 25, 27, 4, 32, 36, 8, 4, 9, 16, 49, 25, 4, 27, 8, 4, 9, 64, 4, 72, 25, 4, 16, 81, 4, 8, 9, 4, 32, 49, 9, 100, 8, 108, 16, 4, 9, 8, 121, 4, 125, 9, 128, 4, 27, 8, 4, 144, 49, 4, 25, 8, 9, 4, 32, 81, 4, 8, 169, 9, 4, 25, 16, 36, 8, 4, 27

Offset: 1

Author

Amiram Eldar, Dec 09 2023

Keywords

• nonn

Comments

The terms of A057521 that are larger than 1, since A057521(k) = 1 if and only if k is squarefree (A005117).

Links

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

Crossrefs

Cf. A005117, A013929, A057521.

Cf. A084936, A174961, A275699, A368038, A368039.

Cf. A013661, A229099, A328013.

Programs

Mathematica

f[p_, e_] := If[e > 1, p^e, 1]; powPart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[powPart, 200], # > 1 &]

PARI

lista(kmax) = {my(p, f); for(k = 1, kmax, f = factor(k); p = prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); if(p > 1, print1(p, ", ")));}

Formula

a(n) = A057521(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = d/(3*(1-1/zeta(2))^(3/2)) = 4.778771..., and d = A328013.