Let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\leq i < k$. Is it true that $v_0(n)=\max_{k\geq 0}v(n,k)\to \infty$ as $n\to \infty$?

Does $V_1(n) = 1$ have finite solutions?

This is a modification of erdos_889.variants.v1_eq_1_finite, which might make it more amenable to attack according to [ErSe67].

[ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.