A308470 - OEIS

%I #31 Sep 08 2022 08:46:21

%S 0,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,1,0,0,2,0,0,0,1,0,1,0,0,1,0,2,0,

%T 1,0,1,1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,1,7,0,0,1,0,1,0,

%U 0,1,0,0,1,1,0,4,7,0,1,0,0,2,0,0,0,1,3,2,0,0,1,0,2,0,4

%N a(n) = (gcd(phi(n), 4*n^2 - 1) - 1)/2, where phi is A000010, Euler's totient function.

%C 2*a(n) + 1 = gcd(phi(2*n), (2*n - 1)*(2*n + 1)).

%C a(A000040(n)) = A099618(n).

%C Records occur at n = 1, 7, 22, 62, 172, 213, 372, 427, 473, ...

%F a(A000040(n)) = A099618(n).

%F a(A002476(n)) = 1.

%F a(A045309(n)) = 0.

%e a(7) = 1 because (gcd(phi(7), 4*7^2 - 1) - 1)/2 = (gcd(6, 195) - 1)/2 = (3 - 1)/2 = 1.

%t Table[(GCD[EulerPhi[n], 4n^2 - 1] - 1)/2, {n, 100}] (* _Alonso del Arte_, May 30 2019 *)

%o (Magma) [(Gcd(EulerPhi(n),4*n^2-1)-1)/2: n in [1..95]];

%o (Python)

%o from fractions import gcd

%o def A000010(n):

%o     if n == 1:

%o         return 1

%o     d, m = 1, 0

%o     while d < n:

%o         if gcd(d,n) == 1:

%o             m = m+1

%o         d = d+1

%o     return m

%o n = 0

%o while n < 30:

%o     n = n+1

%o     print(n,(gcd(A000010(n),4*n**2-1)-1)//2) # _A.H.M. Smeets_, Aug 18 2019

%Y Cf. A000010, A000040, A002476, A045309, A099618.

%K nonn

%O 1,22

%A _Juri-Stepan Gerasimov_, May 29 2019