A390440 - OEIS

%I #8 Nov 06 2025 00:13:25

%S 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,32,33,34,35,37,

%T 38,39,41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,74,

%U 77,78,79,82,83,85,86,87,89,91,93,94,95,96,97,101,102,103,105

%N Exponentially 5-rough numbers: numbers whose prime factorization exponents are all coprime to 6.

%C The asymptotic density of this sequence is d = zeta(6) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^7) = 0.62636565734301010494... .

%C The asymptotic density of the squarefree numbers within this sequence is 1/(zeta(2) * d) = 0.97056263338702466237... .

%H Amiram Eldar, <a href="/A390440/b390440.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n)^s = zeta(6*s) * Product_{p prime} (1 + 1/p^s + 1/p^(5*s) - 1/p^(6*s)).

%t q[n_] := AllTrue[FactorInteger[n][[;; , 2]], CoprimeQ[6, #] &]; Select[Range[105], q]

%o (PARI) isok(k) = vecsum(apply(x -> if(gcd(x, 6) == 1, 0, 1), factor(k)[, 2])) == 0;

%Y Intersection of A268335 and A390437.

%Y Subsequences: A005117, A050997, A092759, A178740, A179664.

%Y Cf. A007310, A013661, A013664.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Nov 05 2025