On the general Smarandache's sigma product of digits

TL;DR

This paper studies the behavior of Smarandache's sigma product of digits sequence, providing bounds, formulas, and insights into related sequences, supported by computational examples.

Contribution

It introduces bounds and closed-form formulas for related sequences, advancing understanding of Smarandache's sigma product of digits.

Findings

Determined upper bounds for sequences related to Smarandache's sigma product

Derived closed-form formulas for sequences a(n) and b(n)

Analyzed the behavior of the general sequence c(n)

Abstract

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.