Sums of perfect powers

Let $P_n$ be the $n$-th perfect power. In this article we obtain asymptotic formulae for the sum $sum^{n}_{i=1}P_i$. We also prove the following formulae[sum^{n}_{i=1}frac{1}{sqrt{P_i}}=log n+C+o(1),qquad sum_{P_nleq x}frac{1}{sqrt{P_n}}=frac{1}{2}log x+C+o(1),]where $C$ is a constant.