On sequences of integers with small prime factors (Q6595599)

The author of this paper considers the difference between consecutive terms in sequences of integers whose greatest prime factor grows slowly. For any integer \(n\), let \(P(n)\) be the greatest prime factor. The following assertion is the main one in the paper.\N\NLet \(y:(0,\infty)\rightarrow [3,\infty)\) be a non-decreasing function, and let \(\{n_1,n_2\ldots\}\) be the increasing sequence of positive integers such that \(P(n_i)\leqslant y(n_i)\). Then, there exists an effectively computable constant \(c\) such that \[\Nn_{i+1}-{n_i}>n_i/(\log n_i)^{\delta(cy(n_{i+1})}, i\geqslant 3, \N\]\Nwhere \N\[\N\delta(x)=\exp\left\{\frac{x\log\log x}{\log x}\right\}.\N\]