A308470 a(n) = (gcd(phi(n), 4*n^2 - 1) - 1)/2, where phi is A000010, Euler's totient function.

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, 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, 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

Offset: 1

Author

Juri-Stepan Gerasimov, May 29 2019

Keywords

• nonn

Comments

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

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

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

Examples

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

Crossrefs

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

Programs

Magma

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

Mathematica

Table[(GCD[EulerPhi[n], 4n^2 - 1] - 1)/2, {n, 100}] (* Alonso del Arte, May 30 2019 *)

Python

from math import gcd
def A000010(n):
    if n == 1:
        return 1
    d, m = 1, 0
    while d < n:
        if gcd(d,n) == 1:
            m = m+1
        d = d+1
    return m
n = 0
while n < 30:
    n = n+1
    print(n,(gcd(A000010(n),4*n**2-1)-1)//2) # A.H.M. Smeets, Aug 18 2019

Formula

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

a(A002476(n)) = 1.

a(A045309(n)) = 0.