Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$.

Wieferich prime is Wieferich number with $n$ prime.

It is an open problem if there are infinitely many Wieferich primes and infinitely many non-Wieferich primes.

In OEIS Wieferich numbers are A077816.

Comment in OEIS claims:

If there are finitely many Wieferich primes (A001220), this sequence is finite.

Let WP denote Wieferich prime and NWN denote non-Wieferich number.

It is unconditionally known that there are infinitely many non-Wieferich numbers with unbounded largest prime factor.

From now on **assume there are only finitely many non-Wieferich primes**
and almost all primes are WP.

Let $p_i$ be distinct WPs. Then $n=\prod p_i$ is Wieferich number at it is not NWN.

Q1 Are there only finitely many squarefree NWNs?

Q2 What other properties NWNs have?

A guess:

Q3 Must the smallest prime factor of the NWNs be bounded?

**Addedd** Initially we asked about the largest prime factor, but Will Sawin
disproved this and Q1 in a comment.

smallestprime factor. $\endgroup$