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

1,2

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

Mathematica

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

Prog

(PARI) isok(k) = k % 2 && if(k == 1, 1, vecmax(factor(k)[, 2]) < 3);
(Python)
from sympy import mobius, integer_nthroot
def A381822(n):
    def f(x): return int(n+x-sum(mobius(k)*(x//k**3+1>>1) for k in range(1, integer_nthroot(x, 3)[0]+1, 2)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 11 2026

Crossrefs

Intersection of A005408 and A004709.

A056911 is a subsequence.

Cf. A002117, A233091.

Sequence in context: A077797 A353076 A033039 * A322660 A390448 A161957

Adjacent sequences: A381819 A381820 A381821 * A381823 A381824 A381825

Keyword

nonn,easy

Author

Amiram Eldar, Mar 08 2025

Status

approved