A390259 Expansion of Sum_{k>=1} (-1)^(k+1)*x^(k^3)/(1-x^(k^3)).

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1

Offset

1,27

Comments

Number of odd cubes dividing n minus number of even cubes dividing n.

Inverse Moebius transform of (-1)^(n+1) * A010057.

First differences of A309082.

Links

Ridouane Oudra, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d^3|n} (-1)^(d+1).

a(n) = A309082(n) - A309082(n-1).

a(n) = 2*A061704(8*n) - 3*A061704(n).

meaning that : if 8|n then a(n) = A061704(n) - 2*A061704(n/8), else a(n) = A061704(n).

Multiplicative with a(2^e) = 1 - floor(e/3), and a(p^e) = 1 + floor(e/3) for p > 2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (3/4)*zeta(3).

Maple

seq(add((-1)^(d+1), d in select(x -> frac(x^(1/3)) = 0, numtheory[divisors](n))), n = 1..120);

Mathematica

CoefficientList[Series[Sum[(-1)^(k+1)*x^(k^3)/(1-x^(k^3)), {k, 1, 120}], {x, 0, 120}], x]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + (2*(f[i, 1]%2)-1)*(f[i, 2]\3)); } \\ Amiram Eldar, Oct 30 2025

Crossrefs

Cf. A000578, A010057, A309082, A061704.

Cf. A048272, A344299, A390260.

Sequence in context: A037879 A178535 A025449 * A047988 A037818 A087116

Adjacent sequences: A390256 A390257 A390258 * A390260 A390261 A390262

Keyword

sign,mult,easy

Author

Ridouane Oudra, Oct 30 2025

Status

approved