RSA: special parameters construction
It is easily seen that the value of $q$ is invariant modulo $e$.  As a consequence, we always have $q \equiv p\cdot (p-1)^{-1} \pmod e$ and thus $q(p-1) \equiv p \pmod{e}$.  So, the resulting modulus $N = pq$ is such that $N \equiv (p+q) \pmod e$.

Example: With $p=10598342506117936057$ and $e=5004898192290387253$, the proposed construction yields the prime $q = 106091356957242239963$.  One can check that $p\cdot q \bmod e = 1577041040681269201 = (p+q) \bmod e$.