Any odd integer except
1 can be written q=2*n+1 with n>0.

Let An be the value of
Euler totient function applied to 2*n+1. E.g.: An = phi(2*n+1).

Cf. OEIS A000010,
A037225 and A363700.

Let Gn be the greatest
common divisor of An and 2*n. E.g.: Gn = gcd(An,2*n).

Noting Rn = An / Gn (Rn
is an integer), we conjecture:

I) If Rn = 1 then
2*n+1 is prime.

II) If Rn > 2 then
2*n+1 is composite.

III) Rn is never equal
to 2 (i.e., An # 2*Gn).

Q) Can you get an
explanation (or proof) of this?