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A372952 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} n/gcd(x_1, x_2, x_3, n).

%I A372952 #25 May 25 2024 09:02:06
%S A372952 1,15,79,239,621,1185,2395,3823,6397,9315,14631,18881,28549,35925,
%T A372952 49059,61167,83505,95955,130303,148419,189205,219465,279819,302017,
%U A372952 388121,428235,518155,572405,707253,735885,923491,978671,1155849,1252575,1487295,1528883
%N A372952 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} n/gcd(x_1, x_2, x_3, n).
%H A372952 Amiram Eldar, <a href="/A372952/b372952.txt">Table of n, a(n) for n = 1..10000</a>
%F A372952 a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_4(d).
%F A372952 From _Amiram Eldar_, May 21 2024: (Start)
%F A372952 Multiplicative with a(p^e) = (p^(4*e+4) - p^(4*e+1) + p - 1)/(p^4-1).
%F A372952 Dirichlet g.f.: zeta(s)*zeta(s-4)/zeta(s-1).
%F A372952 Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(5)/zeta(4) = 0.958057374... . (End)
%F A372952 a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_4(d^2)/sigma_2(d^2). - _Seiichi Manyama_, May 24 2024
%F A372952 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) )^3. - _Seiichi Manyama_, May 25 2024
%t A372952 f[p_, e_] := (p^(4*e+4) - p^(4*e+1) + p - 1)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o A372952 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d,4));
%Y A372952 Column k=3 of A372968.
%Y A372952 Cf. A001159, A008683.
%Y A372952 Cf. A013662, A013663.
%Y A372952 Cf. A371492, A372962.
%K A372952 nonn,mult
%O A372952 1,2
%A A372952 _Seiichi Manyama_, May 18 2024