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

1,1

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

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.

Mathematica

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

Prog

(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, ", "))); }

Crossrefs

Cf. A005117, A013929, A057521.

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

Cf. A013661, A229099, A328013.

Sequence in context: A301639 A123531 A201522 * A117181 A328906 A194623

Adjacent sequences: A368037 A368038 A368039 * A368041 A368042 A368043

Keyword

nonn

Author

Amiram Eldar, Dec 09 2023

Status

approved