%I #16 May 24 2024 17:36:28

%S 1,65,757,4225,16001,49205,119365,271489,554797,1040065,1783541,

%T 3198325,4850977,7758725,12112757,17392769,24211265,36061805,47162485,

%U 67604225,90359305,115930165,148291397,205517173,250266001,315313505,404686153,504317125,595481825

%N a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, n) )^3.

%H Amiram Eldar, <a href="/A371628/b371628.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} phi(n/d) * (n/d)^3 * sigma_6(d^2)/sigma_3(d^2).

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

%F Multiplicative with a(p^e) = (p^(6*e+2)*(p^4+p^3+2*p^2+p+1) - p^(4*e+2)*(p^2-p+1) + p^2+p+1)/((p+1)^2*(p^2+1)*(p^2-p+1)).

%F Dirichlet g.f.: zeta(s)*zeta(s-4)*zeta(s-6)/zeta(s-3)^2.

%F Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(3)*zeta(7)/zeta(4)^2 = 1.034718122... . (End)

%t f[p_, e_] := (p^(6*e+2)*(p^4+p^3+2*p^2+p+1) - p^(4*e+2)*(p^2-p+1) + p^2+p+1)/((p+1)^2*(p^2+1)*(p^2-p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* _Amiram Eldar_, May 24 2024 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^3*sigma(d^2, 6)/sigma(d^2, 3));

%Y Cf. A373059, A371492.

%Y Cf. A372963, A372964.

%Y Cf. A084220.

%Y Cf. A002117, A013662, A013665.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 24 2024