A390260 Expansion of Sum_{k>=1} (-1)^(k+1)*x^(k^4)/(1-x^(k^4)).

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

Offset

1,81

Comments

Number of odd 4th powers dividing n minus number of even 4th powers dividing n.

Inverse Moebius transform of (-1)^(n+1) * A374016.

First differences of A309083.

Links

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

Formula

a(n) = Sum_{d^4|n} (-1)^(d+1).

a(n) = A309083(n) - A309083(n-1).

a(n) = 2*A063775(16*n) - 3*A063775(n).

meaning that : if 16|n then a(n) = A063775(n) - 2*A063775(n/16), else a(n) = A063775(n).

Multiplicative with a(2^e) = 1 - floor(e/4), and a(p^e) = 1 + floor(e/4) for p > 2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 7*zeta(4)/8 = 7*Pi^4/720.

Maple

seq(add((-1)^(d+1), d in select(x -> frac(x^(1/4)) = 0, numtheory[divisors](n))), n = 1..120);

Mathematica

CoefficientList[Series[Sum[(-1)^(k+1)*x^(k^4)/(1-x^(k^4)), {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]\4)); } \\ Amiram Eldar, Oct 30 2025

Crossrefs

Cf. A000583, A374016, A309083, A063775.

Cf. A048272, A344299, A390259.

Sequence in context: A121372 A359432 A338639 * A249351 A123706 A322817

Adjacent sequences: A390257 A390258 A390259 * A390261 A390262 A390263

Keyword

sign,mult,easy

Author

Ridouane Oudra, Oct 30 2025

Status

approved