NOTE: After edit question became about set partitions, which not was I intended, so this is second try.
Is there an algorithm for producing an infinite subset of set of integer partition numbers p(n) such that all elements of this subset are even and we don't need to use some sieve (for example, find all elements of p(n) and then throw out odd). I don't necessarily need all even p(n), just infinite subset of them.
These are Bell numbers $B_n$.
It can be shown that for $n>1$, $B_{n}\equiv B_{n-1}+B_{n-2}\pmod{2}$, which together with initial values implies that $B_{n}\equiv 0\pmod{2}$ iff $n\equiv 2\pmod{3}$.
Namely, consider a mapping on the set partitions of $\{1,2,\dots,n\}$ that exchanges elements $1$ and $2$. This mapping is an involution, which maps a set partition to some other (and vice versa), unless $1$ and $2$ appear in the same part or form their own singleton parts (in which cases the set partition is mapped to itself). The number of such exceptional set partitions of the former type is $B_{n-1}$, and the number of those of the latter type is $B_{n-2}$. This observation implies the congruence $B_{n}\equiv B_{n-1}+B_{n-2}\pmod{2}$.
