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A360429 Inverse Mobius transformation of A034714.

%I A360429 #13 Jan 16 2024 09:04:59
%S A360429 1,9,19,57,51,171,99,313,262,459,243,1083,339,891,969,1593,579,2358,
%T A360429 723,2907,1881,2187,1059,5947,1926,3051,3178,5643,1683,8721,1923,7737,
%U A360429 4617,5211,5049,14934,2739,6507,6441,15963,3363,16929,3699,13851,13362,9531,4419,30267,7302,17334
%N A360429 Inverse Mobius transformation of A034714.
%H A360429 Amiram Eldar, <a href="/A360429/b360429.txt">Table of n, a(n) for n = 1..10000</a>
%F A360429 a(n) = Sum_{d|n} A000005(d)*d^2.
%F A360429 Dirichlet convolution of A034714 and A000012.
%F A360429 Dirichlet g.f.: zeta^2(s-2)*zeta(s).
%F A360429 From _Amiram Eldar_, Feb 09 2023: (Start)
%F A360429 Multiplicative with a(p^e) = ((e+1)*p^(2*e+4) - (e+2)*p^(2*e+2) + 1)/(p^2-1)^2.
%F A360429 Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1/3 + zeta'(3)/zeta(3)) * n^3 * zeta(3)/3, where gamma is Euler's constant (A001620). (End)
%p A360429 A360429 := proc(n)
%p A360429     add(numtheory[tau](d)*d^2,d=numtheory[divisors](n)) ;
%p A360429 end proc:
%t A360429 f[p_, e_] := ((e+1)*p^(2*e+4) - (e+2)*p^(2*e+2) + 1)/(p^2-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 09 2023 *)
%Y A360429 Cf. A000005, A000012, A001620, A034714, A060640.
%K A360429 nonn,mult,easy
%O A360429 1,2
%A A360429 _R. J. Mathar_, Feb 07 2023