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A344139 a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n, x_5|n} gcd(x_1,x_2,x_3,x_4,x_5).

%I A344139 #28 May 12 2021 10:48:22
%S A344139 1,33,34,277,36,1122,38,1335,313,1188,42,9418,44,1254,1224,4771,48,
%T A344139 10329,50,9972,1292,1386,54,45390,391,1452,1720,10526,60,40392,62,
%U A344139 14193,1428,1584,1368,86701,68,1650,1496,48060,72,42636,74,11634,11268,1782,78,162214,477,12903,1632,12188,84
%N A344139 a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n, x_5|n} gcd(x_1,x_2,x_3,x_4,x_5).
%H A344139 Seiichi Manyama, <a href="/A344139/b344139.txt">Table of n, a(n) for n = 1..10000</a>
%F A344139 a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n, x_5|n} n/lcm(x_1,x_2,x_3,x_4,x_5).
%F A344139 a(n) = Sum_{d|n} phi(n/d) * tau(d)^5.
%F A344139 If p is prime, a(p) = 31 + p.
%F A344139 a(n) = Sum_{k=1..n} tau(gcd(k,n))^5.
%t A344139 a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^5 &]; Array[a, 20] (* _Amiram Eldar_, May 10 2021 *)
%o A344139 (PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, sumdiv(n, l, sumdiv(n, m, gcd([i, j, k, l, m]))))));
%o A344139 (PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, sumdiv(n, l, sumdiv(n, m, n/lcm([i, j, k, l, m]))))));
%o A344139 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d)^5);
%o A344139 (PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n))^5);
%Y A344139 Cf. A000203, A060724, A344132, A344138, A344140.
%K A344139 nonn,mult
%O A344139 1,2
%A A344139 _Seiichi Manyama_, May 10 2021