On the general Smarandache's sigma product of digits

This paper investigates the behaviour of one of the most famous Smarandache's sequence given by A061076 on oeis. In particular we first study the behaviour of two sequences (A061077, A061078) strictly connected with the main Smarandache's sigma product of digits. We'll solve some open problems such as the determination of an upper bound for these sequences (which hold for all $n\in\mathbb{N}$) and the determination of a closed formula for each $a(n)$ and $b(n)$. Then combining these results it will be possible to understand the behaviour of the general sequence $c(n)$. Every result will be accompanied by Wolfram Mathematica scripts examples in order to support our thesis.