Primes $p$ such that $p-b$ Has a Large Power Factor and Few Other Prime Divisors

We prove lower bounds for the number of primes $p \leq N + b$ such that $p-b$ is divisible by $2^{k(N)}$ and has at most $k$ odd prime factors ($k \geq 2$), assuming $2^{k(N)} \leq N^\theta$ for some $\theta > 0$ depending on $k$. The proof uses a variant of Chen's method, weighted sieves, and Elliott's results on primes in arithmetic progressions with large power-factor moduli.