A373061 - OEIS

%I #16 May 31 2024 05:49:52

%S 1,9,31,77,141,279,379,637,877,1269,1431,2387,2341,3411,4371,5181,

%T 5169,7893,7183,10857,11749,12879,12651,19747,18041,21069,24043,29183,

%U 25173,39339,30691,41789,44361,46521,53439,67529,51949,64647,72571,89817,70521,105741

%N 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 Amiram Eldar, <a href="/A373061/b373061.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..200 from Seiichi Manyama)

%F From _Amiram Eldar_, May 31 2024: (Start)

%F 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 Dirichlet g.f.: zeta(s)*zeta(s-2)*zeta(s-3)/zeta(s-1)^2.

%F 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 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 (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 Cf. A372952, A373060.

%Y Cf. A002117, A013661, A013662.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 21 2024