For odd integer $n$, define the Euler quotient modulo $n$ to be $a(n)$:

$$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$

$a(n)=0$ for OEIS sequence Wieferich numbers

**Conjecture 1** If a Wieferich prime $p$ divides $n$, then $p$ divides $a(n)$
and in addition $\gcd(n,a(n))>1$.

**Conjecture 2** For $n$ Mersenne number $M_m=2^m-1$ we have
$a(2^m-1)= \phi(2^m-1)/m$.

This holds for $m$ up to 200.

**Conjecture 3** If a Wieferich prime $p$ divides $M_m$, then
$p$ divides $ \phi(2^m-1)/m$.

This holds for $p=1093$.

Which of the conjectures are true?

Question1 Does these conjectures and the answer(s) contribute new results about Wieferich primes?