Almost-primes from an upper-bound sieve and Buchstab identity

For a product of finitely many admissible linear forms, first omit a fixed finite set of locally obstructing primes. Apply the Buchstab identity at a large fixed $w$, and bound each removed term by the Selberg upper-bound sieve. The convergent tail ${\sum }_{p\ge w}1/(p(\log p{)}^k)$ leaves a positive proportion with no prime divisor in $[w,X^{\delta })$. Their distinct prime divisors below $w$ are finite in number, while those above $X^{\delta }$ number at most $O(1/\delta )$, producing infinitely many almost-prime values.