A382891 The powerfree part of the n-th cubefree number.

1, 2, 3, 1, 5, 6, 7, 1, 10, 11, 3, 13, 14, 15, 17, 2, 19, 5, 21, 22, 23, 1, 26, 7, 29, 30, 31, 33, 34, 35, 1, 37, 38, 39, 41, 42, 43, 11, 5, 46, 47, 1, 2, 51, 13, 53, 55, 57, 58, 59, 15, 61, 62, 7, 65, 66, 67, 17, 69, 70, 71, 73, 74, 3, 19, 77, 78, 79, 82, 83

Offset: 1

Author

Amiram Eldar, Apr 07 2025

Keywords

• nonn
• easy

Comments

Also, the squarefree part of the n-th cubefree number.

Links

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

Crossrefs

Cf. A002117, A004709, A005117, A055231, A007913, A371188.

Similar sequences: A382888, A382889, A382890.

Programs

Mathematica

f[p_, e_] := p^If[e == 1, 1, 0]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

PARI

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 1, 0)), ", ")));}

Formula

a(n) = A055231(A004709(n)).

a(n) = A007913(A004709(n)).

a(n) = A004709(n)/A382889(n) = A004709(n)/A382890(n)^2.

a(n) = A382888(n)^2/A004709(n).

a(A371188(n)) = A005117(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^5) = 0.92517253037215590197... .