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%I #8 Oct 21 2023 11:37:03
%S 1,2,3,5,6,7,10,11,13,14,15,16,17,19,21,22,23,26,29,30,31,33,34,35,37,
%T 38,39,41,42,43,46,47,48,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,
%U 74,77,78,79,80,81,82,83,85,86,87,89,91,93,94,95,97,101,102
%N Numbers whose canonical prime factorization contains only exponents which are congruent to 1 modulo 3.
%C First differs from A274034 at n = 42, and from A197680 and A361177 at n = 84.
%C The asymptotic density of this sequence is zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A002117 * A330523 = A253905 * A065465 = 0.644177671086029533405... .
%H Amiram Eldar, <a href="/A366762/b366762.txt">Table of n, a(n) for n = 1..10000</a>
%F Sum_{n>=1} 1/a(n)^s = zeta(3*s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)), for s > 1.
%t q[n_] := AllTrue[FactorInteger[n][[;; , 2]], Mod[#, 3] == 1 &]; Select[Range[120], q]
%o (PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2]%3 != 1, return(0))); 1;}
%Y Similar sequences with exponents of a given form: A000290 (2*k), A268335 (2*k+1), A000578 (3*k), A182120 (3*k+2).
%Y Cf. A002117, A065465, A253905, A330523.
%Y Cf. A197680, A274034, A361177, A366761.
%K nonn,easy
%O 1,2
%A _Amiram Eldar_, Oct 21 2023
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