= 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 $\sum_{p\geq w}1/(p(\log p)^k)$ 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.
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