On the Number of Prime Factors of Consecutive Integers

Abstract

We prove that there are infinitely many $n$ such that $\omega (n+k)\ll \log k$ for all integers $k\ge 2$. This improves on a result of Tao-Teräväinen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress on a number of questions posed by Erdős. The proof is based on a quantitative refinement of the Tao-Teräväinen probabilistic argument, combining a more efficient sieve procedure with stronger exponential concentration-of-measure estimates. Moreover, we formulate a conjecture on integers with many prime factors based on Cramér-type random models. Assuming this conjecture, the main bound is essentially sharp.

32 pages