A326394 Expansion of Sum_{k>=1} x^k * (1 + x^(2*k)) / (1 - x^(3*k)).

1, 1, 2, 2, 1, 3, 2, 2, 3, 2, 1, 5, 2, 2, 3, 3, 1, 5, 2, 3, 4, 2, 1, 6, 2, 2, 4, 4, 1, 6, 2, 3, 3, 2, 2, 8, 2, 2, 4, 4, 1, 6, 2, 3, 5, 2, 1, 8, 3, 3, 3, 4, 1, 7, 2, 4, 4, 2, 1, 9, 2, 2, 6, 4, 2, 6, 2, 3, 3, 4, 1, 10, 2, 2, 5, 4, 2, 6, 2, 5, 5, 2, 1, 10, 2, 2, 3, 4, 1, 10, 4

Offset: 1

Author

Ilya Gutkovskiy, Sep 11 2019

Keywords

• nonn

Comments

Number of divisors of n that are not of the form 3*k + 2.

Crossrefs

Cf. A000005, A001817, A001822, A003627, A032766, A035191, A082051, A326395.

Cf. A001620, A256843.

Programs

Mathematica

nmax = 90; CoefficientList[Series[Sum[x^k (1 + x^(2 k))/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, 1 &, !MemberQ[{2}, Mod[#, 3]] &], {n, 1, 90}]

PARI

a(n) = {numdiv(n) - sumdiv(n, d, d%3==2)} \\ Andrew Howroyd, Sep 11 2019

Formula

a(n) = A000005(n) - A001822(n).

Sum_{k=1..n} a(k) ~ 2*n*log(n)/3 + c*n, where c = (5*gamma-2)/3 - gamma(2,3) = (5*A001620-2)/3 - A256843 = 0.222152... . - Amiram Eldar, Jan 14 2024