The number of shifted almost-primes with smooth values

Let [Formula: see text] and [Formula: see text] denote the largest and smallest prime factors of [Formula: see text], respectively. Let [Formula: see text], [Formula: see text]. For sufficiently large [Formula: see text], we show that there exists a strictly positive constant [Formula: see text] such that [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is fixed, which improves the previous result of Lü-Wang-Chen (2019) with [Formula: see text]. Let [Formula: see text] denote an almost-prime with at most [Formula: see text] prime factors, counted according to multiplicity. In particular, we prove that the pattern [Formula: see text] is realized by a positive proportion of [Formula: see text], for [Formula: see text] and [Formula: see text], which also improves an implicit result of Lü-Wang-Chen (2019) by replacing [Formula: see text] with [Formula: see text].