 Sifting function

For a finite set $A$ of positive integers and a set of prime numbers $\mathcal{P}$, the sifting function counts elements divisible by no prime number in $\mathcal{P}$ strictly below $z$:The strict inequality is the endpoint convention used in the Buchstab identity here. With an empty product, $P(z)=1$ and every element survives.

For a finitely supported nonnegative weighted sequence $\mathcal{A}=(a_n)$, use the same notation for $S(\mathcal{A},\mathcal{P},z)={\sum }_{\gcd (n,P(z))=1}{a}_n$. Choosing $a_n$ to be the indicator function of $A$ recovers the set version. For $A=\{1,\dots ,10\}$, all prime numbers allowed, and $z=5$, only $2$ and $3$ are removed as divisors, leaving $1,5,7$ and giving $S=3$.