A366761 Numbers that have at least one exponent in their canonical prime factorization that is divisible by 3.

8, 24, 27, 40, 54, 56, 64, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 192, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 320, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 432, 440, 448, 456, 459, 472, 488, 500, 504, 512, 513, 520

Offset

1,1

Comments

Each term has a unique representation of as product of two numbers: one is a cube (A000578) and the second is a number that is not in this sequence.

The asymptotic density of this sequence is 1 - zeta(3) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.10483363599014046584... .

From Amiram Eldar, Jan 22 2024: (Start)

The complement of this sequence is the sequence of numbers called "unitarily 3-free", or "3-skew", by Cohen (1961).

He proved that the asymptotic density of unitarily k-free, i.e., numbers whose prime factorization contain no exponent that is divisible by k, is zeta(k) * Product_{p prime} (1 - 2/p^k + 1/p^(k+1)) (see p. 228, eq. 3.18). (End)

Links

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

Eckford Cohen, Some sets of integers related to the k-free integers, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.

Formula

Sum_{n>=1} 1/a(n)^s = zeta(s) * (1 - zeta(3*s) * Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s))), for s > 1.

Mathematica

q[n_] := ! AllTrue[FactorInteger[n][[;; , 2]], ! Divisible[#, 3] &]; Select[Range[500], q]

Prog

(PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(!(f[i, 2]%3), return(1))); 0; }

Crossrefs

A000578 is a subsequence.

Cf. A002117, A182120, A366762.

Sequence in context: A301517 A374459 A195086 * A336593 A176297 A375072

Adjacent sequences: A366758 A366759 A366760 * A366762 A366763 A366764

Keyword

nonn,easy

Author

Amiram Eldar, Oct 21 2023

Status

approved