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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.

The weighted sifting function isLet ${\lambda }_1=1$ and let the real Selberg sieve weights ${\lambda }_d$ vanish unless $d\mid P_z$ and $d\le D$. Sinceexpansion and the distribution hypothesis giveFor the optimizing Selberg weights, the main quadratic form is $G(D,z{)}^{-1}$, whereThus the general upper bound isIf the sieve level is instead defined as the largest possible least common multiple, one supports the individual weights on $d\le \sqrt{D}$; this is the same statement after replacing $D$ by $\sqrt{D}$.

Take $\mathcal{P}$ to contain the primes at least $5$; the exceptional local behavior at $2$ and $3$ is then harmless. Fix a sufficiently large constant $w$ and later choosewith fixed small $0<\delta <\eta$. The Buchstab identity givesThe first term is $c_{w}X+O_w(1)$ for a positive constant $c_w$, by direct counting in the finitely many permitted residue classes modulo $P_w$.

For each term in the sum, part c supplies the local factor $g(p)=3/p$ and remainders bounded by powers of $3^{\omega (d)}$. Apply the Selberg upper-bound sieve to the remaining prime conditions. Mertens theorem gives the dimension-three densityso the main terms in the Buchstab sum are bounded byThis convergent tail can be made smaller than $c_{w}X/4$ by taking $w$ large. The weighted remainder terms are $o(X)$: the estimate $k^{\omega (n)}{\ll }_{k,\epsilon }{n}^{\epsilon }$ controls the summed remainders, while part d controls uniformly the loss caused by the finite sieve level. Choosing $\delta$ sufficiently small relative to $\eta$, and then taking $X$ large, therefore givesfor some absolute $c>0$.

For every counted $m$, the distinct prime divisors of $F(m)$ are either below the fixed $w$ or at least $X^{\delta }$. The first class contains at most $\pi (w)$ primes, while $F(m)\ll X^3$ makes the second class contain at mostprimes. Since every prime divisor of $m$, $m+2$, or $m+6$ divides $F(m)$,after enlarging an absolute constant $C$. A positive proportion occurs for arbitrarily large $X$, so infinitely many such $m$ exist. This is the almost-primes from an upper-bound sieve and Buchstab identity method.

Replace $F(m)$ by $m(m+1)(m+2)$. The only change is at the locally obstructing primes: every value is divisible by $2$, and every value is divisible by $3$, so omit both primes from $\mathcal{P}$. For every $p\ge 5$, the three roots $0,-1,-2$ are distinct and again give $g(p)=3/p$ and $r_d=O(3^{\omega (d)})$. All Selberg upper-bound sieve, Buchstab identity, and large-prime-factor estimates from part e are unchanged, proving the analogous result.