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A373061 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, n).

%I A373061 #16 May 31 2024 05:49:52
%S A373061 1,9,31,77,141,279,379,637,877,1269,1431,2387,2341,3411,4371,5181,
%T A373061 5169,7893,7183,10857,11749,12879,12651,19747,18041,21069,24043,29183,
%U A373061 25173,39339,30691,41789,44361,46521,53439,67529,51949,64647,72571,89817,70521,105741
%N A373061 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, n).
%H A373061 Amiram Eldar, <a href="/A373061/b373061.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..200 from Seiichi Manyama)
%F A373061 From _Amiram Eldar_, May 31 2024: (Start)
%F A373061 Multiplicative with a(p^e) = (p^(3*e)*(p+1)^3 - p^(2*e)*(p^2+p+1) + 1)/((p^2+p+1)*(p+1)).
%F A373061 Dirichlet g.f.: zeta(s)*zeta(s-2)*zeta(s-3)/zeta(s-1)^2.
%F A373061 Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(2) * zeta(4) / zeta(3)^2 = Pi^6/(540*zeta(3)^2) = 1.232126852811... . (End)
%t A373061 f[p_, e_] := (p^(3*e)*(p+1)^3 - p^(2*e)*(p^2+p+1) + 1)/((p^2+p+1)*(p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 31 2024 *)
%o A373061 (PARI) a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, gcd([i, j, n])/gcd([i, j, k, n]))));
%Y A373061 Cf. A372952, A373060.
%Y A373061 Cf. A002117, A013661, A013662.
%K A373061 nonn,mult
%O A373061 1,2
%A A373061 _Seiichi Manyama_, May 21 2024