Sum of perfect powers of n natural numbers

Question

The perfect power of an integer $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}$ are $p_1^2, p_1^3\cdots p_1^{\alpha_1},p_2^2, p_2^3 \cdots p_2^{\alpha_2},(p_1 p_2)^2, (p_2 p_3)^2\cdots$ similar terms.

Lets define $S(n)$ as the sum of all perfect powers that can be formed from prime factors of n.

Examples: \begin{align} S(9) &= 1+9 = 10 \\ S(16) &= 1+4+8+16 = 29 \\ S(36) &= 1+4+9+36 = 50 \end{align}

Let $$G(n) = \sum_{i=1}^n S(n)$$ is there a short closed form of the sum $G(n)$?

References.

1. M. A. Nyblom, A counting function for the sequence of perfect powers, Austral. Math. Soc. Gaz. 33 (2006), 338–343.

2. R. Jakimczuk, On the distribution of perfect powers, J. Integer Seq. 14 (2011), Article 11.8.5.

3. similar problem