A358714 - OEIS

%I #24 Mar 10 2023 12:19:45

%S 1,1,8,8,64,8,216,64,216,64,1000,64,1728,216,512,512,4096,216,5832,

%T 512,1728,1000,10648,512,8000,1728,5832,1728,21952,512,27000,4096,

%U 8000,4096,13824,1728,46656,5832,13824,4096,64000,1728,74088,8000,13824,10648,97336,4096,74088

%N a(n) = phi(n)^3.

%C Number of solutions to gcd(x*y*z, n) = 1 such that 0 <= x,y,z <= n-1.

%C x*y*z == t (mod n) where t is a unit (invertible element) in Z_n. Since t is a unit, all x,y,z must be units. Here there are A000010(n) possibilities for each x,y,z so there are a total of A000010(n)^3 ways to get t as a unit.

%F a(n) = A000010(n)^3.

%F From _Amiram Eldar_, Jan 06 2023: (Start)

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

%F Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime}(1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.08429696844... .

%F Sum_{k>=1} 1/a(k) = Product_{p prime} (1 + p^3/((p-1)^3*(p^3-1))) = 2.47619474816... (A335818). (End)

%e a(9) = A000010(9)^3 = 216.

%t a[n_] := EulerPhi[n]^3; Array[a, 100] (* _Amiram Eldar_, Jan 06 2023 *)

%o (Magma)  [(EulerPhi(n))^3: n in [1..180]];

%o (PARI) a(n) = eulerphi(n)^3;

%Y Cf. A000010, A127473, A361148, A335818.

%K nonn,easy,mult

%O 1,3

%A _Param Mayurkumar Parekh_ and _Paavan Mayurkumar Parekh_, Nov 28 2022