A347176 G.f.: Sum_{k>=1} (-1)^(k+1) * k * x^(k^2) / (1 - x^(k^2)).

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, -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, 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

Offset: 1

Author

Ilya Gutkovskiy, Aug 21 2021

Keywords

• sign
• mult

Comments

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

nmax = 90; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) #^(1/2) &, IntegerQ[#^(1/2)] &], {n, 1, 90}]
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 *)

PARI

a(n) = sumdiv(n, d, if (issquare(d), (-1)^((d%2)+1)*sqrtint(d))); \\ Michel Marcus, Aug 22 2021

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

Formula

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

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