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A344132 a(n) = Sum_{i|n, j|n, k|n} gcd(i,j,k).

%I A344132 #26 May 12 2021 10:16:49
%S A344132 1,9,10,37,12,90,14,111,49,108,18,370,20,126,120,283,24,441,26,444,
%T A344132 140,162,30,1110,79,180,184,518,36,1080,38,657,180,216,168,1813,44,
%U A344132 234,200,1332,48,1260,50,666,588,270,54,2830,117,711,240,740,60,1656,216,1554,260,324,66,4440,68,342,686,1441,240,1620,74
%N A344132 a(n) = Sum_{i|n, j|n, k|n} gcd(i,j,k).
%H A344132 Seiichi Manyama, <a href="/A344132/b344132.txt">Table of n, a(n) for n = 1..10000</a>
%F A344132 a(n) = Sum_{i|n, j|n, k|n} n/lcm(i,j,k).
%F A344132 a(n) = Sum_{d|n} phi(n/d) * tau(d)^3.
%F A344132 If p is prime, a(p) = 7 + p.
%F A344132 a(n) = Sum_{k=1..n} tau(gcd(k,n))^3.
%t A344132 a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^3 &]; Array[a, 50] (* _Amiram Eldar_, May 10 2021 *)
%o A344132 (PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, gcd([i, j, k]))));
%o A344132 (PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, n/lcm([i, j, k]))));
%o A344132 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d)^3);
%o A344132 (PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n))^3);
%Y A344132 Cf. A060724, A344133, A344134, A344135, A344138, A344139, A344140.
%K A344132 nonn,mult
%O A344132 1,2
%A A344132 _Seiichi Manyama_, May 10 2021