A366762 Numbers whose canonical prime factorization contains only exponents which are congruent to 1 modulo 3.

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, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset

1,2

Comments

First differs from A274034 at n = 42, and from A197680 and A361177 at n = 84.

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... .

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^(3*s)), for s > 1.

Mathematica

q[n_] := AllTrue[FactorInteger[n][[;; , 2]], Mod[#, 3] == 1 &]; Select[Range[120], q]

Prog

(PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2]%3 != 1, return(0))); 1; }

Crossrefs

Similar sequences with exponents of a given form: A000290 (2*k), A268335 (2*k+1), A000578 (3*k), A182120 (3*k+2).

Cf. A002117, A065465, A253905, A330523.

Cf. A197680, A274034, A361177, A366761.

Sequence in context: A274034 A197680 A361177 * A369210 A369937 A119024

Adjacent sequences: A366759 A366760 A366761 * A366763 A366764 A366765

Keyword

nonn,easy

Author

Amiram Eldar, Oct 21 2023

Status

approved