%I #19 May 24 2024 17:36:10

%S 1,17,91,289,701,1547,2647,4769,7705,11917,15731,26299,30421,44999,

%T 63791,77473,87857,130985,136459,202589,240877,267427,290951,433979,

%U 448201,517157,633187,764983,729989,1084447,951391,1248929,1431521,1493569,1855547,2226745

%N a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^2.

%H Amiram Eldar, <a href="/A371492/b371492.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_4(d^2)/sigma_2(d^2).

%F From _Amiram Eldar_, May 24 2024: (Start)

%F Multiplicative with a(p^e) = (p^(4*e+1)*(p+1)*(p^2+p+1) - p^(3*e+1)*(p^2+1) + p + 1)/((p^2+1)*(p^2+p+1)).

%F Dirichlet g.f.: zeta(s)*zeta(s-3)*zeta(s-4)/zeta(s-2)^2.

%F Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2)*zeta(5)/zeta(3)^2 = 1.180448217... . (End)

%t f[p_, e_] := (p^(4*e+1)*(p+1)*(p^2+p+1) - p^(3*e+1)*(p^2+1) + p + 1)/((p^2+1)*(p^2+p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 36] (* _Amiram Eldar_, May 24 2024 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^2*sigma(d^2, 4)/sigma(d^2, 2));

%Y Cf. A084218, A373060.

%Y Cf. A372952, A372962.

%Y Cf. A373059, A371628.

%Y Cf. A002117, A013661, A013663.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 24 2024