A347176 - OEIS

%I #19 Mar 01 2023 02:02:16

%S 1,1,1,-1,1,1,1,-1,4,1,1,-1,1,1,1,-5,1,4,1,-1,1,1,1,-1,6,1,4,-1,1,1,1,

%T -5,1,1,1,-4,1,1,1,-1,1,1,1,-1,4,1,1,-5,8,6,1,-1,1,4,1,-1,1,1,1,-1,1,

%U 1,4,-13,1,1,1,-1,1,1,1,-4,1,1,6,-1,1,1,1,-5,13,1,1,-1,1,1,1,-1,1,4

%N G.f.: Sum_{k>=1} (-1)^(k+1) * k * x^(k^2) / (1 - x^(k^2)).

%C Excess of sum of square roots of odd square divisors of n over sum of square roots of even square divisors of n.

%H Amiram Eldar, <a href="/A347176/b347176.txt">Table of n, a(n) for n = 1..10000</a>

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

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2) (A002162). - _Amiram Eldar_, Mar 01 2023

%t nmax = 90; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[DivisorSum[n, (-1)^(# + 1) #^(1/2) &, IntegerQ[#^(1/2)] &], {n, 1, 90}]

%t f[p_, e_] := (p^(Floor[e/2] + 1) - 1)/(p - 1); f[2, e_] := 3 - 2^(Floor[e/2] + 1); 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)*sqrtint(d))); \\ _Michel Marcus_, Aug 22 2021

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

%Y Cf. A002129, A002162, A037213, A069290, A344299, A344300.

%K sign,mult

%O 1,9

%A _Ilya Gutkovskiy_, Aug 21 2021