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

%I A366761 #13 Jan 22 2024 01:27:14
%S A366761 8,24,27,40,54,56,64,72,88,104,108,120,125,135,136,152,168,184,189,
%T A366761 192,200,216,232,248,250,264,270,280,296,297,312,320,328,343,344,351,
%U A366761 360,375,376,378,392,408,424,432,440,448,456,459,472,488,500,504,512,513,520
%N A366761 Numbers that have at least one exponent in their canonical prime factorization that is divisible by 3.
%C A366761 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.
%C A366761 The asymptotic density of this sequence is 1 - zeta(3) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.10483363599014046584... .
%C A366761 From _Amiram Eldar_, Jan 22 2024: (Start)
%C A366761 The complement of this sequence is the sequence of numbers called "unitarily 3-free", or "3-skew", by Cohen (1961).
%C A366761 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)
%H A366761 Amiram Eldar, <a href="/A366761/b366761.txt">Table of n, a(n) for n = 1..10000</a>
%H A366761 Eckford Cohen, <a href="http://pub.acta.hu/acta/showCustomerArticle.action?id=7038&amp;dataObjectType=article&amp;returnAction=showCustomerVolume&amp;sessionDataSetId=3fe5dd60a67f39f9">Some sets of integers related to the k-free integers</a>, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.
%F A366761 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.
%t A366761 q[n_] := ! AllTrue[FactorInteger[n][[;; , 2]], ! Divisible[#, 3] &]; Select[Range[500], q]
%o A366761 (PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(!(f[i, 2]%3), return(1))); 0;}
%Y A366761 A000578 is a subsequence.
%Y A366761 Cf. A002117, A182120, A366762.
%K A366761 nonn,easy
%O A366761 1,1
%A A366761 _Amiram Eldar_, Oct 21 2023