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

Author

Amiram Eldar, Oct 21 2023

Keywords

• nonn
• easy

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.

Crossrefs

A000578 is a subsequence.

Cf. A002117, A182120, A366762.

Programs

Mathematica

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

PARI

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

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.