A349692 - OEIS

%I #12 Dec 05 2021 06:12:36

%S 1,6,10,25,18,60,26,88,67,108,42,250,50,156,180,280,66,402,74,450,260,

%T 252,90,880,211,300,372,650,114,1080,122,832,420,396,468,1675,146,444,

%U 500,1584,162,1560,170,1050,1206,540,186,2800,435,1266,660,1250,210,2232,756

%N Dirichlet convolution of the gcd-sum function (A018804) with itself.

%H Vaclav Kotesovec, <a href="/A349692/a349692.jpg">Asymptotics of Sum_{k=1..n} a(k) with a graph</a>

%F Dirichlet g.f.: zeta(s-1)^4 / zeta(s)^2.

%F a(n) = Sum_{d|n} A018804(d) * A018804(n/d).

%F a(n) = Sum_{d|n} A029935(d) * A038040(n/d).

%F a(n) = Sum_{d|n} A344683(d) * A000203(n/d).

%F Multiplicative with a(p^e) = (e+1) * p^(e-2) * ((e+2)*(e+3)*p^2 - 2*e*(e+2)*p + e*(e-1))/6. - _Amiram Eldar_, Nov 25 2021

%t A018804[n_] := Sum[GCD[n,k], {k, 1, n}]; a[n_] := Sum[A018804[d] A018804[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]

%t f[p_, e_] := (e + 1)*p^(e - 2)*((e + 2)*(e + 3)*p^2 - 2*e*(e + 2)*p + e*(e - 1))/6; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* _Amiram Eldar_, Nov 25 2021 *)

%o (PARI)

%o A029935(n) = sumdiv(n, d, eulerphi(d)*eulerphi(n/d)); \\ From A029935.

%o A349692(n) = sumdiv(n, d, A029935(n/d)*d*numdiv(d)); \\ _Antti Karttunen_, Nov 25 2021

%Y Cf. A000203, A018804, A029935, A038040, A344683.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Nov 25 2021