On the number of distinct prime factors of $$nj+a^hk$$ n j + a h k

Abstract

Let $$\omega (n)$$ denote the number of distinct prime factors of $$n$$ . Then for any given $$K\ge 2$$ , small $$\epsilon >0$$ and sufficiently large (only depending on $$K$$ and $$\epsilon $$ ) $$x$$ , there exist at least $$x^{1-\epsilon }$$ integers $$n\in [x,(1+K^{-1})x]$$ such that $$\omega (nj\pm a^hk)\ge (\log \log \log x)^{\frac{1}{3}-\epsilon }$$ for all $$2\le a\le K$$ , $$1\le j,k\le K$$ and $$0\le h\le K\log x$$ .