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Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-2) / (zeta(s-1))^2.
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%I #10 Oct 16 2022 08:00:04

%S 1,1,4,5,16,4,36,21,40,16,100,20,144,36,64,85,256,40,324,80,144,100,

%T 484,84,416,144,364,180,784,64,900,341,400,256,576,200,1296,324,576,

%U 336,1600,144,1764,500,640,484,2116,340,1800,416,1024,720,2704,364,1600,756,1296,784

%N Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-2) / (zeta(s-1))^2.

%F Multiplicative with a(1) = 1 and a(p^e) = (p^(2*e)-1) * (p-1) / (p+1) for prime p and e > 0.

%F Dirichlet convolution of A002618 and A023900.

%F Dirichlet convolution of A001157 and A328722.

%F Dirichlet inverse b(n) for n > 0 is multiplicative with b(1) = 1 and b(p^e) = -(p-1)^2 * e * p^(e-1) for prime p and e > 0.

%F Dirichlet convolution with A060640 equals A007433.

%F Dirichlet convolution with A018804 equals A000290.

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = 12*zeta(3)/Pi^4 = 0.148083... . - _Amiram Eldar_, Oct 16 2022

%t f[p_, e_] := (p^(2*e) - 1)*(p - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 24 2021 *)

%Y Cf. A000290, A001157, A002618, A007433, A018804, A023900, A060640, A328722.

%K nonn,easy,mult

%O 1,3

%A _Werner Schulte_, Jan 24 2021