A332844 - OEIS

%I #13 Feb 07 2023 08:28:34

%S 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 42,44,64,30,72,32,81,48,54,48,112,38,60,56,108,42,96,44,96,84,72,48,

%U 156,58,96,72,112,54,132,72,144,80,90,60,192,62,96,112,166,84

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

%H Antti Karttunen, <a href="/A332844/b332844.txt">Table of n, a(n) for n = 1..20000</a>

%F G.f.: Sum_{k>=1} sigma(k) * (theta_3(x^k) - 1) / 2.

%F a(n) = Sum_{d|n} A076752(d).

%F a(n) = Sum_{d|n} A206369(n/d) * tau(d).

%F a(n) = Sum_{d|n} A010052(n/d) * sigma(d).

%F a(n) = Sum_{d|n} A124315(n/d) * phi(d).

%F a(n) = Sum_{d|n} A046951(n/d) * d.

%F a(p) = p + 1, where p is prime.

%F Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 1080. - _Vaclav Kotesovec_, Feb 26 2020

%t Table[Sum[Boole[IntegerQ[(n/d)^(1/2)]] DivisorSigma[1, d], {d, Divisors[n]}], {n, 1, 65}]

%t nmax = 65; CoefficientList[Series[Sum[DivisorSigma[1, k] (EllipticTheta[3, 0, x^k] - 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%o (PARI) A332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d)); \\ _Antti Karttunen_, May 23 2021

%Y Cf. A000005, A000010, A000203, A010052, A046951, A076752 (Mobius transf.), A124315, A206369, A344442, A347090 (Dirichlet inverse).

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Feb 26 2020