A326395 Expansion of Sum_{k>=1} x^(2*k) * (1 + x^k) / (1 - x^(3*k)).

0, 1, 1, 1, 1, 3, 0, 2, 2, 2, 1, 4, 0, 2, 3, 2, 1, 5, 0, 3, 2, 2, 1, 6, 1, 2, 3, 2, 1, 6, 0, 3, 3, 2, 2, 7, 0, 2, 2, 4, 1, 6, 0, 3, 5, 2, 1, 7, 0, 3, 3, 2, 1, 7, 2, 4, 2, 2, 1, 9, 0, 2, 4, 3, 2, 6, 0, 3, 3, 4, 1, 10, 0, 2, 4, 2, 2, 6, 0, 5, 4, 2, 1, 8, 2, 2, 3, 4, 1, 10

Offset

1,6

Comments

Number of divisors of n that are not of the form 3*k + 1.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000005(n) - A001817(n).

Sum_{k=1..n} a(k) ~ 2*n*log(n)/3 + c*n, where c = (5*gamma-2)/3 - gamma(1,3) = (5*A001620-2)/3 - A256425 = -0.382447... . - Amiram Eldar, Jan 14 2024

Maple

N:= 100: # for a(1) .. a(N)
S:= series(add(x^(2*k)*(1+x^k)/(1-x^(3*k)), k=1..N/2), x, N+1):
seq(coeff(S, x, i), i=1..N); # Robert Israel, Aug 27 2020

Mathematica

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

Prog

(PARI) a(n) = {numdiv(n) - sumdiv(n, d, d%3==1)} \\ Andrew Howroyd, Sep 11 2019

Crossrefs

Cf. A000005, A001817, A001822, A004611 (positions of 0's), A007494, A035191, A082050, A326394.

Cf. A001620, A256425.

Sequence in context: A176314 A004587 A104609 * A290566 A285006 A369747

Adjacent sequences: A326392 A326393 A326394 * A326396 A326397 A326398

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 11 2019

Status

approved