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A338997 Number of (i,j,k) in {1,2,...,n}^3 such that gcd(n,i) = gcd(n,j) = gcd(n,k).

%I A338997 #51 Nov 15 2022 09:17:19
%S A338997 1,2,9,10,65,18,217,74,225,130,1001,90,1729,434,585,586,4097,450,5833,
%T A338997 650,1953,2002,10649,666,8065,3458,6057,2170,21953,1170,27001,4682,
%U A338997 9009,8194,14105,2250,46657,11666,15561,4810,64001,3906,74089,10010,14625,21298,97337,5274,74305,16130
%N A338997 Number of (i,j,k) in {1,2,...,n}^3 such that gcd(n,i) = gcd(n,j) = gcd(n,k).
%H A338997 Amiram Eldar, <a href="/A338997/b338997.txt">Table of n, a(n) for n = 1..10000</a>
%F A338997 a(n) = Sum_{d|n} phi(d)^3.
%F A338997 From _Seiichi Manyama_, Mar 13 2021: (Start)
%F A338997 a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^2.
%F A338997 G.f.: Sum_{k>=1} phi(k)^3 * x^k/(1 - x^k). (End)
%F A338997 From _Amiram Eldar_, Nov 15 2022: (Start)
%F A338997 Multiplicative with a(p^e) = 1 + ((p-1)^2 (p^(3*e)-1))/(p^2 + p + 1).
%F A338997 Sum_{k=1..n} a(k) ~ c * n^4, where c = (Pi^4/360) * Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.09123656748... . (End)
%t A338997 a[n_] := DivisorSum[n, EulerPhi[#]^3 &]; Array[a, 100] (* _Amiram Eldar_, Dec 31 2020 *)
%o A338997 (PARI) a(n)=sumdiv(n,d,eulerphi(d)^3)
%o A338997 (PARI) a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^2); \\ _Seiichi Manyama_, Mar 13 2021
%o A338997 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^3*x^k/(1-x^k))) \\ _Seiichi Manyama_, Mar 13 2021
%Y A338997 Cf. A000010, A029939.
%K A338997 nonn,mult
%O A338997 1,2
%A A338997 _Benoit Cloitre_, Dec 31 2020