A390441 Numbers whose prime factorization exponents are neither 2 nor 3.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 96, 97

Offset

1,2

Comments

First differs from its subsequences A252895 and A366242 at n = 43: a(43) = 64 is neither a term in A252895 nor a term in A366242.

The squarefree numbers (A005117) and the 4-full numbers (A036967) are all terms in this sequence. Each term in this sequence has a unique representation as the product of two coprime numbers, one is squarefree and the other is 4-full.

The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^2 + 1/p^4) = 0.66922021803510257394... .

Links

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

Formula

Sum_{n>=1} 1/a(n)^s = zeta(s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(4*s)).

Mathematica

q[n_] := AllTrue[FactorInteger[n][[;; , 2]], !MemberQ[{2, 3}, #] &]; Select[Range[100], q]

Prog

(PARI) isok(k) = vecsum(apply(x -> if(x == 2 || x == 3, 1, 0), factor(k)[, 2])) == 0;

Crossrefs

Intersection of A268335 and A386799.

Subsequences: A005117, A036967, A252895, A366242.

Sequence in context: A318239 A059266 A336222 * A252895 A366242 A336224

Adjacent sequences: A390438 A390439 A390440 * A390442 A390443 A390444

Keyword

nonn,easy

Author

Amiram Eldar, Nov 05 2025

Status

approved