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A344223 a(n) = Sum_{k=1..n} tau(gcd(k,n)^n), where tau(n) is the number of divisors of n.

%I A344223 #22 May 16 2021 08:06:44
%S A344223 1,4,6,16,10,72,14,64,45,180,22,600,26,336,360,256,34,1620,38,1600,
%T A344223 672,792,46,4752,175,1092,378,3080,58,36960,62,1024,1584,1836,1680,
%U A344223 17136,74,2280,2184,12960,82,97020,86,7480,9450,3312,94,37536,441,16900,3672,10400,106,40824,3960,25200
%N A344223 a(n) = Sum_{k=1..n} tau(gcd(k,n)^n), where tau(n) is the number of divisors of n.
%H A344223 Seiichi Manyama, <a href="/A344223/b344223.txt">Table of n, a(n) for n = 1..10000</a>
%F A344223 a(n) = n * A344226(n).
%F A344223 a(n) = Sum_{d|n} phi(n/d) * tau(d^n).
%F A344223 a(n) = n * Sum_{d|n} n^omega(d) / d.
%F A344223 If p is prime, a(p) = 2*p.
%t A344223 Table[Sum[DivisorSigma[0,GCD[k,n]^n],{k,n}],{n,100}] (* _Giorgos Kalogeropoulos_, May 13 2021 *)
%o A344223 (PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n)^n));
%o A344223 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^n));
%o A344223 (PARI) a(n) = n*sumdiv(n, d, n^omega(d)/d);
%Y A344223 Cf. A000203, A332517, A344221, A344222, A344224, A344225, A344226.
%K A344223 nonn
%O A344223 1,2
%A A344223 _Seiichi Manyama_, May 12 2021