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Egyptian Fractions and Prime Power Divisors
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John Machacek

Department of Mathematics

Michigan State University

619 Red Cedar Road

C212 Wells Hall

East Lansing, MI 48824

USA

**Abstract:**

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.

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(Concerned with sequences
A000668
A003306
A005835
A007850
A019434
A054377
A073932
A073935
A283423
A286497
A286499.)

Received June 7 2017; revised versions received September 14 2017; January 5 2018; January 22 2018; March 15 2018.
Published in *Journal of Integer Sequences*, March 29 2018.

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