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A371492 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^2.

%I A371492 #19 May 24 2024 17:36:10
%S A371492 1,17,91,289,701,1547,2647,4769,7705,11917,15731,26299,30421,44999,
%T A371492 63791,77473,87857,130985,136459,202589,240877,267427,290951,433979,
%U A371492 448201,517157,633187,764983,729989,1084447,951391,1248929,1431521,1493569,1855547,2226745
%N A371492 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 A371492 Amiram Eldar, <a href="/A371492/b371492.txt">Table of n, a(n) for n = 1..10000</a>
%F A371492 a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_4(d^2)/sigma_2(d^2).
%F A371492 From _Amiram Eldar_, May 24 2024: (Start)
%F A371492 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 A371492 Dirichlet g.f.: zeta(s)*zeta(s-3)*zeta(s-4)/zeta(s-2)^2.
%F A371492 Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2)*zeta(5)/zeta(3)^2 = 1.180448217... . (End)
%t A371492 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 A371492 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^2*sigma(d^2, 4)/sigma(d^2, 2));
%Y A371492 Cf. A084218, A373060.
%Y A371492 Cf. A372952, A372962.
%Y A371492 Cf. A373059, A371628.
%Y A371492 Cf. A002117, A013661, A013663.
%K A371492 nonn,mult
%O A371492 1,2
%A A371492 _Seiichi Manyama_, May 24 2024