Correlations of almost primes

Abstract We prove that analogues of the Hardy–Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula number integers $n=p_1p_2 \leq X$ such $n+h$ is a product exactly two which holds all $|h|\leq H$ with $\log^{19+\varepsilon}X\leq H\leq X^{1-\varepsilon}$ , under restriction size one factors n and . Additionally, we consider correlations $n,n+h$ where has factors, establishing $|h| $X^{1/6+\varepsilon}\leq