Let $p_1<\cdots <p_u$ be primes and let $k\ge 1$. Let $F_k(p_1,\dots ,p_u)$ be such that every interval of $k$ positive integers contains at least $F_k(p_1,\dots ,p_u)$ multiples of at least one of the $p_i$.Estimate $F_k(p_1,\dots ,p_u)$, particularly in the range $k=\alpha p_u$ for constant $\alpha >2$.