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%I #20 Jan 06 2023 05:41:56
%S 1,1,4,2,6,4,8,4,13,6,12,8,14,8,24,8,18,13,20,12,32,12,24,16,31,14,40,
%T 16,30,24,32,16,48,18,48,26,38,20,56,24,42,32,44,24,78,24,48,32,57,31,
%U 72,28,54,40,72,32,80,30,60,48,62,32,104,32,84,48
%N Expansion of Sum_{k>=1} k * x^k * (1 - x^k + x^(2*k)) / (1 - x^(4*k)).
%H Robert Israel, <a href="/A327095/b327095.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: Sum_{k>=1} x^k * (1 + 3 * x^(2*k) + x^(3*k) + 3 * x^(4*k) + x^(6*k)) / (1 - x^(4*k))^2.
%F a(n) = Sum_{d|n, n/d odd} d - Sum_{d|n, n/d twice odd} d.
%F a(n) = A002131(n) if n odd, A002131(n) - A002131(n/2) if n even.
%F From _Amiram Eldar_, Dec 05 2022: (Start)
%F Multiplicative with a(2^e) = 2^(e-1), and a(p^e) = (p^(e+1)-1)/(p-1) if p > 2.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/64 = 0.462637... . (End)
%F Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/2^s)^2. - _Amiram Eldar_, Jan 06 2023
%p N:= 100:
%p G:= add(k * x^k * (1 - x^k + x^(2*k)) / (1 - x^(4*k)),k=1..N):
%p S:= series(G,x,N+1):
%p [seq(coeff(S,x,i),i=1..N)];# _Robert Israel_, Sep 17 2019
%t nmax = 66; CoefficientList[Series[Sum[k x^k (1 - x^k + x^(2 k))/(1 - x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A002131[n_] := Total[Select[Divisors[n], OddQ[n/#] &]]; a[n_] := If[OddQ[n], A002131[n], A002131[n] - A002131[n/2]]; Table[a[n], {n, 1, 66}]
%t f[p_, e_] := (p^(e + 1) - 1)/(p - 1); f[2, e_] := 2^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 05 2022 *)
%o (PARI) a(n)={sumdiv(n, d, d*((n/d%2==1) - (n/d%4==2)))} \\ _Andrew Howroyd_, Sep 13 2019
%Y Cf. A002131, A115607, A285895, A316631 (Moebius transform).
%K nonn,mult
%O 1,3
%A _Ilya Gutkovskiy_, Sep 13 2019
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