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%I #19 Feb 28 2023 03:39:54
%S 1,5,5,8,5,25,5,11,8,25,5,40,5,25,25,14,5,40,5,40,25,25,5,55,8,25,11,
%T 40,5,125,5,17,25,25,25,64,5,25,25,55,5,125,5,40,40,25,5,70,8,40,25,
%U 40,5,55,25,55,25,25,5,200,5,25,40,20,25,125,5,40,25,125
%N Multiplicative with a(p^e) = 3*e + 2.
%H Vaclav Kotesovec, <a href="/A360909/b360909.txt">Table of n, a(n) for n = 1..10000</a>
%F Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 3/p^s - 1/p^(2*s)).
%F Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)), (with a product that converges for s=1).
%F Sum_{k=1..n} a(k) ~ c * n * log(n)^4 / 24, where c = Product_{primes p} (1 - 7/p^2 + 11/p^3 - 6/p^4 + 1/p^5) = 0.091414252314317101861531055690354339957600046..., more precise (but very complicated) asymptotics can be obtained (in Mathematica notation) as Residue[Zeta[s]^5 * f[s] * n^s / s, {s, 1}], where f[s] = Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)).
%t a[n_] := Times @@ ((3*Last[#] + 2) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Feb 25 2023 *)
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1+3*X-X^2)/(1-X)^2)[n], ", "))
%Y Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), this sequence (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).
%K nonn,mult
%O 1,2
%A _Vaclav Kotesovec_, Feb 25 2023
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