On a sumset problem for integers

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 $|A|\geq 5$.

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