A382888 The squarefree kernel of the n-th cubefree number.

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

Offset: 1

Author

Amiram Eldar, Apr 07 2025

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A002117, A004709, A005117, A007947, A371188.

Similar sequences: A382889, A382890, A382891.

Programs

Mathematica

s[n_] := Module[{f = FactorInteger[n]}, If[AllTrue[f[[;; , 2]], # < 3 &], Times @@ f[[;; , 1]], Nothing]]; Array[s, 100]

PARI

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(vecprod(f[, 1]), ", ")));}

Formula

a(n) = A007947(A004709(n)).

a(n) = A004709(n)/sqrt(A382889(n)) = A004709(n)/A382890(n).

a(n) = sqrt(A004709(n)*A382891(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^3 - 1/p^4) = 0.98875409459226057523... .