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

1,2

Comments

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

Links

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

Formula

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

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A004709, A005117.

Cf. A253905, A306633.

Sequence in context: A101886 A284509 A135674 * A195121 A047332 A248233

Adjacent sequences: A371185 A371186 A371187 * A371189 A371190 A371191

Keyword

nonn,easy

Author

Amiram Eldar, Mar 14 2024

Status

approved