Might be related to Wieferich primes.

Let $p$ be odd prime and define the Fermat quotient

$$F(n)=\frac{(2^{n-1} -1)}{n} \mod n=\frac{(2^{n-1} \bmod n^2 )-1}{n}$$

For integer $b$ let $L_p(b)$ be the $p$-adic logarithm with precision two and let $a(p)=\frac{L_p(2)}{p}$.

$a(p)$ is integer in $[0,p-1]$ and in pari/gp the logarithm is defined as log(b+O(p^2)) and in sage Qp(p,2)(b).log()

Q1 Are there infinitely many primes for which $L_p(2)$ and $a(p)$ are zero?

Q2 Are there infinitely many primes for which $L_p(2)$ and $a(p)$ are non-zero?

Q3 If $p$ is Wieferich prime is $L_p(2)=0$?

Q4 If $p$ is non-Wieferich do we have $p=F(p)+a(p)$? this holds for up to 10^7

/*
pari/gp code for p-adic logarithm and Fermat quotient
*/
{L(p)=lift(log(2+O(p^2)))}
{a(n)=L(n)/n;}
{F(n)=lift(Mod(2,n^2)^(n-1)-1)/n}

{f(L=10^2)=
forprime(p=3,L,
b=F(p)+a(p);
if(b!=p,print([p,b,F(p),a(p)]));
)};

f(4000)
```