The sum of a prime power and an almost prime

For any fixed $k\ge 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take $M(k)=(2+\epsilon )k$ for any $\epsilon >0$, or $M(k)=(1+\epsilon )k$ under the assumption of the Elliott–Halberstam conjecture. Moreover, we give a variant of this result which accounts for congruence conditions and strengthens a classical theorem of Erdős and Rao. The main tools we employ are the weighted sieve method of Diamond, Halberstam and Richert, bounds on the number of representations of an integer as the sum of two $k$th powers, and results on $k$th power residues. We also use some simple computations and arguments to conjecture an optimal value of $M(k)$, as well as a related variant of Hardy and Littlewood’s Conjecture H.