Let us consider the function $\sigma:\mathbb{N}\to\mathbb{N}$ defined as:
If $\prod_{i=1}^{r}p_i^{\alpha_i}$ is the prime-factorization of $n$, then $$ \sigma(n)=\sum_{i=1}^{r}\alpha_ip_i $$ So in some sense, $\sigma$ is an "integer-logarithm". Then we define $\iota(n)$ to be the number of iterations of $\sigma$ until we obtain a fix point of sigma (i.e. prime or 4, i think these are the only ones). So for example $\iota(8)=2$. Finally, we consider the sequence $(a_i)_{i\in\mathbb{N}}\subset\mathbb{N}$ such that $\iota(a_i)=i$ and $\iota(n)<i$ for all $1<n<a_i$. The first few terms are (I wrote a little program): $$2,6,8,14,26,62,134,393,1257,4659,9314,27933,167073,334142$$ What can we say about the rate of growth of $(a_i)$? Something curious seems to be that $a_{i+1}$ is always surprisingly close to an integer multiple of $a_i$. Can this be explained in some sort?
This question is purely recreational and just out of fun, but I would like to know if one can say something interesting about this sequence.