(1)-INITIAL IDEAS:
Playing, like always, with my computer, I´ve been plotting this function, that includes the Euler totient function, and the Divisor function.
$latex \displaystyle f(n)=Mod( \phi(n),\sigma_0(n))\: , \; n\in\mathbb{N_{*}^{+}}$
The first 25 values of $latex \displaystyle f(n)$, (not in OEIS) are:
$latex \displaystyle \{0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 2,...\}$
Applying $latex \displaystyle f(x)$ to the Fermat Numbers, $latex \displaystyle F_n=2^{2^{n}}+1$, and to the Mersenne Numbers, $latex \displaystyle M_n=2^n-1$, we can conjecture the following congruences:
(1) $latex \displaystyle \phi(F_n) \equiv 0\; (mod \; \sigma_0(F_n))$
(2) $latex \displaystyle \phi(F_n-2) \equiv 0\; (mod \; \sigma_0(F_n-2))$
(3) $latex \displaystyle \phi(M_n) \equiv 0\; (mod \; \sigma_0(M_n))$
(4) $latex \displaystyle \phi(M_n+2) \equiv 0\; (mod \; \sigma_0(M_n+2))$
¿How do they can be proved? [1]
Archives:
[a]-020809-FERMAT AND MERSENNE NUMBER CONJECTURE-(1).nb
References:
[1]-My Math Forum/Number Theory: Mersenne and Fermat Numbers congruence
