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A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).

%I A373059 #19 May 27 2024 07:14:58
%S A373059 1,5,13,25,41,65,85,121,157,205,221,325,313,425,533,569,545,785,685,
%T A373059 1025,1105,1105,1013,1573,1441,1565,1777,2125,1625,2665,1861,2617,
%U A373059 2873,2725,3485,3925,2665,3425,4069,4961,3281,5525,3613,5525,6437,5065,4325,7397,5965
%N A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).
%H A373059 Seiichi Manyama, <a href="/A373059/b373059.txt">Table of n, a(n) for n = 1..1000</a>
%F A373059 a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_2(d^2)/sigma(d^2).
%F A373059 From _Amiram Eldar_, May 27 2024: (Start)
%F A373059 Multiplicative with a(p^e) = (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2.
%F A373059 Dirichlet g.f.: zeta(s) * zeta(s-2)^2 / zeta(s-1)^2.
%F A373059 Sum_{k=1..n} a(k) ~ (2*zeta(3)*n^3/(15*zeta(4))) * (log(n) + 2*gamma - 1/3 - 2*zeta'(2)/zeta(2) + zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)
%t A373059 f[p_, e_] := (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 27 2024 *)
%o A373059 (PARI) a(n) = sum(i=1, n, sum(j=1, n, gcd(i, n)/gcd([i, j, n])));
%o A373059 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2);} \\ _Amiram Eldar_, May 27 2024
%Y A373059 Row sums of A350900.
%Y A373059 Cf. A057660, A350156.
%Y A373059 Cf. A001620, A002117, A013661, A073002, A244115, A306016.
%K A373059 nonn,mult
%O A373059 1,2
%A A373059 _Seiichi Manyama_, May 21 2024