 Solution

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.