A390260 - OEIS

%I #11 Nov 01 2025 15:07:43

%S 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,

%T 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,

%U 1,1,1,1,1,1,1,1,1,1,1,0,2,1,1,1,1,1,1,1,1,1

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

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

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

%C First differences of A309083.

%H Ridouane Oudra, <a href="/A390260/b390260.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

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

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

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

%t CoefficientList[Series[Sum[(-1)^(k+1)*x^(k^4)/(1-x^(k^4)), {k, 1, 120}], {x, 0, 120}], x]

%o (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

%Y Cf. A000583, A374016, A309083, A063775.

%Y Cf. A048272, A344299, A390259.

%K sign,mult,easy

%O 1,81

%A _Ridouane Oudra_, Oct 30 2025