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A344138 a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n} gcd(x_1,x_2,x_3,x_4).

%I A344138 #23 May 12 2021 10:17:02
%S A344138 1,17,18,99,20,306,22,373,119,340,26,1782,28,374,360,1115,32,2023,34,
%T A344138 1980,396,442,38,6714,165,476,532,2178,44,6120,46,2901,468,544,440,
%U A344138 11781,52,578,504,7460,56,6732,58,2574,2380,646,62,20070,219,2805,576,2772,68,9044,520,8206,612,748,74,35640,76
%N A344138 a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n} gcd(x_1,x_2,x_3,x_4).
%H A344138 Seiichi Manyama, <a href="/A344138/b344138.txt">Table of n, a(n) for n = 1..10000</a>
%F A344138 a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n} n/lcm(x_1,x_2,x_3,x_4).
%F A344138 a(n) = Sum_{d|n} phi(n/d) * tau(d)^4.
%F A344138 If p is prime, a(p) = 15 + p.
%F A344138 a(n) = Sum_{k=1..n} tau(gcd(k,n))^4.
%t A344138 a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^4 &];  Array[a, 50] (* _Amiram Eldar_, May 10 2021 *)
%o A344138 (PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, sumdiv(n, l, gcd([i, j, k, l])))));
%o A344138 (PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, sumdiv(n, l, n/lcm([i, j, k, l])))));
%o A344138 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d)^4);
%o A344138 (PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n))^4);
%Y A344138 Cf. A344132, A344139, A344140.
%K A344138 nonn,mult
%O A344138 1,2
%A A344138 _Seiichi Manyama_, May 10 2021