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A332844 Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s).

%I A332844 #17 May 26 2025 09:54:42
%S A332844 1,3,4,8,6,12,8,18,14,18,12,32,14,24,24,39,18,42,20,48,32,36,24,72,32,
%T A332844 42,44,64,30,72,32,81,48,54,48,112,38,60,56,108,42,96,44,96,84,72,48,
%U A332844 156,58,96,72,112,54,132,72,144,80,90,60,192,62,96,112,166,84
%N A332844 Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s).
%H A332844 Antti Karttunen, <a href="/A332844/b332844.txt">Table of n, a(n) for n = 1..20000</a>
%F A332844 G.f.: Sum_{k>=1} sigma(k) * (theta_3(x^k) - 1) / 2.
%F A332844 a(n) = Sum_{d|n} A076752(d).
%F A332844 a(n) = Sum_{d|n} A206369(n/d) * tau(d).
%F A332844 a(n) = Sum_{d|n} A010052(n/d) * sigma(d).
%F A332844 a(n) = Sum_{d|n} A124315(n/d) * phi(d).
%F A332844 a(n) = Sum_{d|n} A046951(n/d) * d.
%F A332844 a(p) = p + 1, where p is prime.
%F A332844 Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 1080. - _Vaclav Kotesovec_, Feb 26 2020
%F A332844 Multiplicative with a(p^e) = (2*p^(e+3) - e*p^2 + e - 3*p^2 + 1)/(2*(p-1)*(p^2-1)) if e is odd, and (2*p^(e+3) - e*p^2 + e - 2*p^2 - 2*p + 2)/(2*(p-1)*(p^2-1)) if e is even. - _Amiram Eldar_, May 25 2025
%t A332844 Table[Sum[Boole[IntegerQ[(n/d)^(1/2)]] DivisorSigma[1, d], {d, Divisors[n]}], {n, 1, 65}]
%t A332844 nmax = 65; CoefficientList[Series[Sum[DivisorSigma[1, k] (EllipticTheta[3, 0, x^k] - 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A332844 f[p_, e_] := (2*p^(e + 3) - e*p^2 + e - If[OddQ[e], 3*p^2 - 1, 2*p^2 + 2*p - 2])/(2*(p - 1)*(p^2 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* _Amiram Eldar_, May 25 2025 *)
%o A332844 (PARI) A332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d)); \\ _Antti Karttunen_, May 23 2021
%Y A332844 Cf. A000005, A000010, A000203, A010052, A046951, A076752 (Mobius transf.), A124315, A206369, A344442, A347090 (Dirichlet inverse).
%K A332844 nonn,mult
%O A332844 1,2
%A A332844 _Ilya Gutkovskiy_, Feb 26 2020