Recall that a *Wieferich prime* is a prime number $p$ such that
$2^{p-1} \equiv 1 \bmod p^2.$
It is not known whether there are infinitely many Wieferich primes, nor whether there are infinitely many non-Wieferich primes. In fact there are only $2$ known Wieferich primes.

I'm interested in a slightly different condition which I'm hoping is easier to handle. Namely, I can replace the exponent $p-1$ by the order of $2$ in $(\mathbb{Z}/p\mathbb{Z})^\times$. Moreover, I just want this power to hit the identity with odd $p$-adic valuation. Specifically:

Are there infinitely many primes $p$ such that $v_p(2^{\mathrm{ord}_p(2)}-1)$ is odd?

Here $\mathrm{ord}_p(n)$ denotes the order of $n$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ (which divides $p-1$ by FLT), and $v_p$ is the $p$-adic valuation.

Note that the existence of infinitely many non-Wieferich primes would provide a positive answer to my question (since here $v_p(2^{p-1}-1) = 1$).

Ideally I'd also like to know that there collection of such primes has positive density, rather than just being infinite.

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