A390440 Exponentially 5-rough numbers: numbers whose prime factorization exponents are all coprime to 6.

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, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 96, 97, 101, 102, 103, 105

Offset

1,2

Comments

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... .

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

Links

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

Formula

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)).

Mathematica

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

Prog

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

Crossrefs

Intersection of A268335 and A390437.

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

Cf. A007310, A013661, A013664.

Sequence in context: A382913 A240370 A193304 * A394256 A333634 A348499

Adjacent sequences: A390437 A390438 A390439 * A390441 A390442 A390443

Keyword

nonn,easy

Author

Amiram Eldar, Nov 05 2025

Status

approved