  Let $A$ be a finite set of integers. We show that if $k$ is a prime power or
a product of two distinct primes then $$|A+k\cdot A|\geq(k+1)|A|-\lceil
k(k+2)/4\rceil$$ provided $|A|\geq (k-1)^{2}k!$, where $A+k\cdot A=\{a+kb:\
a,b\in A\}$. We also establish the inequality $|A+4\cdot A|\geq 5|A|-6 $ for
$|A|\geq 5$.
