A383008 -id:A383008 - cp's OEIS frontend

A383009 Indices of the even terms in the sequence of cubefree numbers.

2, 4, 6, 9, 11, 13, 16, 18, 20, 23, 24, 26, 29, 31, 33, 36, 38, 40, 43, 45, 49, 51, 53, 56, 58, 60, 63, 65, 67, 69, 71, 73, 76, 78, 80, 83, 85, 87, 90, 93, 96, 98, 100, 103, 105, 106, 109, 111, 113, 115, 117, 119, 122, 124, 126, 129, 131, 133, 137, 139, 142, 144

Offset: 1

Amiram Eldar, Apr 12 2025

The asymptotic density of this sequence is 3/7.

In general, the asymptotic density of the indices of the even terms in the sequence of k-free numbers (numbers that are not divisible by a k-th power other than 1), for k >= 2, is (2^(k-1)-1)/(2^k-1).

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

Cf. A004709, A381822, A383004, A383008.

cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; Position[Select[Range[250], cubeFreeQ], _?EvenQ] // Flatten

iscubefree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 2, return (0))); 1; }
list(lim) = {my(c = 0); for(k = 1, lim, if(iscubefree(k), c++; if(!(k % 2), print1(c, ", ")))); }

A383004(a(n)) > 0.

cp's OEIS Frontend

A383009 Indices of the even terms in the sequence of cubefree numbers.

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