A344300 - OEIS

%I #15 Nov 15 2022 09:17:28

%S 1,1,1,-3,1,1,1,-3,10,1,1,-3,1,1,1,-19,1,10,1,-3,1,1,1,-3,26,1,10,-3,

%T 1,1,1,-19,1,1,1,-30,1,1,1,-3,1,1,1,-3,10,1,1,-19,50,26,1,-3,1,10,1,

%U -3,1,1,1,-3,1,1,10,-83,1,1,1,-3,1,1,1,-30,1,1,26,-3,1,1,1,-19

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

%C Excess of sum of odd squares dividing n over sum of even squares dividing n.

%H Amiram Eldar, <a href="/A344300/b344300.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(2^e) = 2 - (2^(2*floor(e/2) + 2) - 1)/3, and a(p^e) = (p^(2*floor(e/2) + 2) - 1)/(p^2 - 1) for p > 2. - _Amiram Eldar_, Nov 15 2022

%t nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[DivisorSum[n, (-1)^(# + 1) # &, IntegerQ[#^(1/2)] &], {n, 1, 80}]

%t f[p_, e_] := (p^(2*Floor[e/2] + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*Floor[e/2] + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 15 2022 *)

%o (PARI) a(n) = sumdiv(n, d, if (issquare(d), (-1)^((d%2)+1)*d)); \\ _Michel Marcus_, Aug 22 2021

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1]==2, 2 - (2^(2*floor(f[i,2]/2) + 2) - 1)/3, (f[i,1]^(2*floor(f[i,2]/2) + 2) - 1)/(f[i,1]^2 - 1)));} \\ _Amiram Eldar_, Nov 15 2022

%Y Cf. A002129, A035316, A300853, A321543, A344299.

%K sign,mult

%O 1,4

%A _Ilya Gutkovskiy_, May 14 2021