From varying Egyptian fraction equations, we obtain generalizations of primary pseudoperfect numbers and Giuga numbers, which we call prime power pseudoperfect numbers and prime power Giuga numbers, respectively. We show that a sequence of Murthy in the On-line Encyclopedia of Integer Sequences is a subsequence of the sequence of prime power pseudoperfect numbers. We also provide prime factorization conditions sufficient to imply that a number is a prime power pseudoperfect number or a prime power Giuga number. The conditions on prime factorizations naturally give rise to a generalization of Fermat primes, which we call extended Fermat primes.