Journal of Integer Sequences, Vol. 21 (2018), Article 18.3.7

Egyptian Fractions and Prime Power Divisors

John Machacek
Department of Mathematics
Michigan State University
619 Red Cedar Road
C212 Wells Hall
East Lansing, MI 48824


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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