  For each $m\geq 1$, there exist infinitely many primes
$p_1<p_2<\ldots<p_{m+1}$ such that $p_{m+1}-p_1=O(m^4e^{8m})$ and $p_j+2$ has
at most $\frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37$ prime divisors for each
$j$.
