New Representations of Perfect Numbers Using Divisor Functions

The purpose of this paper is to develop a method
to analyze prime factors of perfect numbers,
applicable to both odd and even perfect numbers.
The aim is to elucidate the relationship between the prime factors
of a given perfect number.
It is shown that any perfect number has at least
two distinct prime factors and their divisor properties
are described.  A novel method is proposed
to construct a perfect number, assuming the existence
of $k$-perfect numbers under certain conditions.

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