A390437 Numbers whose prime factorization exponents are not divisible by 3.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset

1,2

Comments

Subsequence of A386799 and first differs from it at n = 58: A386799(58) = 64 = 2^6 is not a term in this sequence.

First differs from A336592 at n = 58: A336592(58) = 64 = 2^6 is not a term in this sequence. Also, a(115) = 128 = 2^7 is the least term that is not a term in A336592.

A182120 and A366762 are subsequences. Each term in this sequence has a unique representation as the product of two coprime numbers, one in A182120 and the other in A366762.

Numbers whose unitary divisors (A077610) that are larger than 1 are all noncubes (A007412).

The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p*(1+p+p^2))) = 0.95692470821076622881... .

Links

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

Formula

Sum_{n>=1} 1/a(n)^s = zeta(3*s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s) - 1/p^(3*s)) for s > 1.

Mathematica

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

Prog

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

Crossrefs

Subsequence of A386799.

Subsequences: A001651, A005117, A004709, A182120, A366762.

Cf. A007412, A077610.

Numbers without exponents that are divisible by m: A268335 (m = 2), this sequence (m = 3), A390438 (m = 4).

Sequence in context: A270420 A336592 A386799 * A392632 A335275 A337052

Adjacent sequences: A390434 A390435 A390436 * A390438 A390439 A390440

Keyword

nonn,easy

Author

Amiram Eldar, Nov 05 2025

Status

approved