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A361265 Multiplicative with a(p^e) = e * p^(e + 1), e > 0.

%I A361265 #17 Sep 01 2023 04:09:37
%S A361265 1,4,9,16,25,36,49,48,54,100,121,144,169,196,225,128,289,216,361,400,
%T A361265 441,484,529,432,250,676,243,784,841,900,961,320,1089,1156,1225,864,
%U A361265 1369,1444,1521,1200,1681,1764,1849,1936,1350,2116,2209,1152,686,1000,2601,2704
%N A361265 Multiplicative with a(p^e) = e * p^(e + 1), e > 0.
%H A361265 Amiram Eldar, <a href="/A361265/b361265.txt">Table of n, a(n) for n = 1..10000</a>
%F A361265 Dirichlet g.f.: Product_{primes p} (1 + p^(s + 2)/(p^s - p)^2).
%F A361265 Dirichlet g.f.: zeta(s-2) * zeta(s-1)^2 * Product_{primes p} (1 - p^(4 - 3*s) + p^(2 - 2*s) + 2*p^(3 - 2*s) - p^(4 - 2*s) - 2*p^(1 - s)).
%F A361265 Sum_{k=1..n} a(k) ~ c * Pi^4 * n^3 / 108, where c = Product_{primes p} (1 - 3/p^2 + 2/p^3 + 1/p^4 - 1/p^5) = 0.3086489554825164955853322259998244718829914385...
%F A361265 a(n) = A005361(n) * A064549(n).
%F A361265 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - log(1-1/p))/p = 1.6843597117... . - _Amiram Eldar_, Sep 01 2023
%t A361265 g[p_, e_] := e*p^(e+1); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
%o A361265 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1 + p^2 * X / (1 - p*X)^2)[n], ", "))
%Y A361265 Cf. A005361, A064549, A203639, A361268.
%K A361265 nonn,easy,mult
%O A361265 1,2
%A A361265 _Vaclav Kotesovec_, Mar 06 2023