# Egyptian Fractions and Prime Power Divisors

## TL;DR

This paper introduces new classes of numbers called prime power pseudoperfect and Giuga numbers, generalizing known concepts through Egyptian fractions and prime factorization conditions, and explores their properties and connections to extended Fermat primes.

## Contribution

It defines prime power pseudoperfect and Giuga numbers, establishes their properties, and introduces extended Fermat primes as a new generalization of Fermat primes.

## Key findings

- Sequence of Murthy is a subsequence of prime power pseudoperfect numbers
- Prime factorization conditions characterize these numbers
- Introduction of extended Fermat primes as a new concept

## Abstract

From varying Egyptian fraction equations we obtain generalizations of primary pseudoperfect numbers and Giuga numbers which we call prime power psuedoperfect numbers and prime power Giuga numbers respectively. We show that a sequence of Amarnath Murthy in the OEIS is a subsequence of the sequence of prime power psuedoperfect numbers. Prime factorization conditions sufficient to imply a number is a prime power pseudoperfect number or a prime power Giuga number are given. The conditions on prime factorizations naturally give rise to a generalization of Fermat primes which we call extended Fermat primes.