A381822 Odd cubefree numbers: odd numbers that are not divisible by any cube greater than 1.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 127, 129, 131

Offset: 1

Author

Amiram Eldar, Mar 08 2025

Keywords

• nonn
• easy

Comments

Numbers whose prime factorization has only odd primes, and all its exponents are smaller than 3 (except for 1 whose prime factorization is empty).

The asymptotic density of this sequence is 4/(7*zeta(3)) = 1/(2*A233091) = 0.475375641474689982104... .

In general, the asymptotic density of odd k-free numbers (numbers that are not divisible by a k-th power other than 1, k >= 2) is 2^(k-1)/((2^k-1) * zeta(k)).

Links

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

Crossrefs

Intersection of A005408 and A004709.

A056911 is a subsequence.

Cf. A002117, A233091.

Programs

Mathematica

cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;;, 2]], # < 3 &]; Select[Range[1, 150, 2], cubeFreeQ]

PARI

isok(k) = k % 2 && if(k == 1, 1, vecmax(factor(k)[, 2]) < 3);