A371188 Indices of the squarefree numbers in the sequence of cubefree numbers.

1, 2, 3, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 19, 20, 21, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 40, 41, 44, 46, 47, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 72, 73, 74, 75, 77, 79, 80, 81, 82, 86, 87, 88, 89, 90, 91

Offset: 1

Author

Amiram Eldar, Mar 14 2024

Keywords

• nonn
• easy

Comments

The asymptotic density of this sequence is zeta(3)/zeta(2) = 0.730762... (A253905).

Examples

			The first 5 cubefree numbers are 1, 2, 3, 4, and 5. The 1st, 2nd, 3rd, and 5th, 1, 2, 3, and 5, are squarefree. Therefore, the first 4 terms of this sequence are 1, 2, 3, and 5.
		

Links

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

Crossrefs

Cf. A004709, A005117.

Cf. A253905, A306633.

Programs

Mathematica

freeQ[n_, k_] := AllTrue[FactorInteger[n], Last[#] < k &]; Position[Select[Range[200], freeQ[#, 3] &], _?(freeQ[#1, 2] &), Heads -> False] // Flatten

PARI

isfree(n, k) = n == 1 || vecmax(factor(n)[, 2]) < k;
lista(kmax) = {my(c = 0); for(k = 1, kmax, if(isfree(k, 3), c++; if(isfree(k, 2), print1(c, ", "))));}

Formula

A004709(a(n)) = A005117(n).

a(n) ~ c * n, where c = zeta(2)/zeta(3) = 1.368432... (A306633).