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A349469 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1)*zeta(s-3)/(zeta(s-2))^2.

%I A349469 #16 Nov 02 2024 03:56:53
%S A349469 1,2,12,20,80,24,252,168,360,160,1100,240,1872,504,960,1360,4352,720,
%T A349469 6156,1600,3024,2200,11132,2016,10400,3744,9828,5040,22736,1920,27900,
%U A349469 10912,13200,8704,20160,7200,47952,12312,22464,13440,65600,6048,75852,22000,28800,22264,99452,16320,88200,20800
%N A349469 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1)*zeta(s-3)/(zeta(s-2))^2.
%H A349469 Vaclav Kotesovec, <a href="/A349469/b349469.txt">Table of n, a(n) for n = 1..10000</a>
%F A349469 Multiplicative with a(p^e) = p^e * (p^(2*e)-1) * (p-1) / (p+1) for e > 0 and prime p.
%F A349469 Dirichlet convolution with A057660 equals A068963.
%F A349469 Equals n * A340850(n) for n > 0.
%F A349469 Dirichlet inverse b(n) for n > 0 is multiplicative with b(1) = 1 and
%F A349469   b(p^e) = -(p-1)^2 * e * p^(2*e-1) for prime p and e > 0.
%F A349469 Sum_{k=1..n} a(k) ~ c * n^4, where c = 9*zeta(3)/Pi^4 = 0.111062... . - _Amiram Eldar_, Oct 16 2022
%t A349469 f[p_, e_] := (p - 1)*p^e*(p^(2*e) - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 18 2021 *)
%Y A349469 Cf. A057660, A068963, A340850.
%K A349469 nonn,easy,mult
%O A349469 1,2
%A A349469 _Werner Schulte_, Nov 18 2021