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

1,22

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, ...

Links

Table of n, a(n) for n=1..95.

Formula

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

a(A002476(n)) = 1.

a(A045309(n)) = 0.

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A085983 A285701 A088183 * A070140 A361561 A081212

Adjacent sequences: A308467 A308468 A308469 * A308471 A308472 A308473

Keyword

nonn

Author

Juri-Stepan Gerasimov, May 29 2019

Status

approved