A338997 - OEIS

%I #51 Nov 15 2022 09:17:19

%S 1,2,9,10,65,18,217,74,225,130,1001,90,1729,434,585,586,4097,450,5833,

%T 650,1953,2002,10649,666,8065,3458,6057,2170,21953,1170,27001,4682,

%U 9009,8194,14105,2250,46657,11666,15561,4810,64001,3906,74089,10010,14625,21298,97337,5274,74305,16130

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

%H Amiram Eldar, <a href="/A338997/b338997.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} phi(d)^3.

%F From _Seiichi Manyama_, Mar 13 2021: (Start)

%F a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^2.

%F G.f.: Sum_{k>=1} phi(k)^3 * x^k/(1 - x^k). (End)

%F From _Amiram Eldar_, Nov 15 2022: (Start)

%F Multiplicative with a(p^e) = 1 + ((p-1)^2 (p^(3*e)-1))/(p^2 + p + 1).

%F 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 a[n_] := DivisorSum[n, EulerPhi[#]^3 &]; Array[a, 100] (* _Amiram Eldar_, Dec 31 2020 *)

%o (PARI) a(n)=sumdiv(n,d,eulerphi(d)^3)

%o (PARI) a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^2); \\ _Seiichi Manyama_, Mar 13 2021

%o (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 Cf. A000010, A029939.

%K nonn,mult

%O 1,2

%A _Benoit Cloitre_, Dec 31 2020