Odd Spoof Multiperfect Numbers

TL;DR

This paper extends the concept of spoof perfect numbers to multiperfect numbers, discovering new odd examples and proposing an algorithm for finding such numbers, thereby advancing understanding of their properties.

Contribution

It introduces the notion of odd spoof multiperfect numbers, presents new examples, and discusses an algorithm for their search, expanding the study of multiperfect numbers.

Findings

Identified new odd spoof multiperfect numbers similar to Descartes' number

Proposed an algorithm for searching for such numbers

Discussed properties of odd spoof multiperfect numbers

Abstract

We generalize the definition of spoof perfect numbers to multiperfect numbers and study their characteristics. As a result, we find several new odd spoof multiperfect numbers, akin to Descartes' number. An example is $8999757$, which would be an odd multiperfect number, if one of its prime factors, $61$, was a square. We briefly describe an algorithm for searching for such numbers and discuss a few of their properties.