OPEN
Are there infinitely many $n$ such that if
\[n(n+1) = \prod_i p_i^{k_i}\]
is the factorisation into distinct primes then all exponents $k_i$ are distinct?
It is likely that there are infinitely many primes $p$ such that $8p^2-1$ is also prime, in which case this is true with exponents $\{1,2,3\}$, letting $n=8p^2$.
