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A309324 Expansion of Sum_{k>=1} psi(k) * x^k/(1 + x^k), where psi = Dedekind psi function (A001615).

%I A309324 #16 Nov 06 2022 03:16:55
%S A309324 1,2,5,2,7,10,9,2,17,14,13,10,15,18,35,2,19,34,21,14,45,26,25,10,37,
%T A309324 30,53,18,31,70,33,2,65,38,63,34,39,42,75,14,43,90,45,26,119,50,49,10,
%U A309324 65,74,95,30,55,106,91,18,105,62,61,70,63,66,153,2,105,130,69,38,125,126,73
%N A309324 Expansion of Sum_{k>=1} psi(k) * x^k/(1 + x^k), where psi = Dedekind psi function (A001615).
%C A309324 Dirichlet convolution of sum of odd divisors function with characteristic function of squarefree numbers.
%H A309324 Amiram Eldar, <a href="/A309324/b309324.txt">Table of n, a(n) for n = 1..10000</a>
%F A309324 a(n) = Sum_{d|n} (-1)^(n/d+1) * psi(d).
%F A309324 a(n) = Sum_{d|n} mu(n/d)^2 * A000593(d).
%F A309324 Multiplicative with a(2^e) = 2, and a(p^e) = (p^e*(p+1)-2)/(p-1) for odd primes p. - _Amiram Eldar_, Dec 01 2020
%F A309324 Sum_{k=1..n} a(k) ~ (5/8) * n^2. - _Amiram Eldar_, Nov 06 2022
%t A309324 nmax = 71; CoefficientList[Series[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k]  x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A309324 Table[Sum[MoebiusMu[n/d]^2 Plus @@ Select[Divisors@ d, OddQ], {d, Divisors[n]}], {n, 1, 71}]
%t A309324 f[2, e_] := 2; f[p_, e_] := (p^e*(p+1)-2)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 01 2020 *)
%o A309324 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 2, (f[i,1]^f[i,2]*(f[i,1]+1)-2)/(f[i,1]-1)));} \\ _Amiram Eldar_, Nov 06 2022
%Y A309324 Cf. A000593, A001615, A008966, A060648, A193356.
%K A309324 nonn,mult
%O A309324 1,2
%A A309324 _Ilya Gutkovskiy_, Jul 23 2019