In mathematics, and in particular number theory, a primary pseudoperfect number is a number N that satisfies the Egyptian fraction equation Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... A modern definition of an Egyptian fraction says that a sum of distinct unit fractions (that is, fractions whose numerators are equal to 1) whose denominators are positive integers, and all of whose denominators differ from each other. ...

where the sum is over only the prime divisors of N. Equivalently (as can be seen by multiplying this equation by N),

Except for the exceptional primary pseudoperfect number 2, this expression gives a representation for N as a sum of a set of distinct divisors of N; therefore each such number (except 2) is pseudoperfect. In mathematics, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. ...

Primary pseudoperfect numbers were first investigated and named by Butske, Jaje, and Mayernik (2000). The first few primary pseudoperfect numbers are

2, 6, 42, 1806, 47058, 2214502422, 52495396602, ... (sequence A054377 in OEIS).

The first four of these numbers are one less than the corresponding numbers in Sylvester's sequence but later numbers in Sylvester's sequence do not similarly correspond to primary pseudoperfect numbers. It is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any odd primary pseudoperfect numbers. 2 (two) is a number, numeral, and glyph. ... Number 6, in this article, refers to the mathematical number. ... Look up forty-two in Wiktionary, the free dictionary. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... Sylvesters sequence consists of the coprime denominators of an Egyptian fraction that adds up to 1. ...

See also Giuga number. A Giuga number is a composite number n such that each of its distinct prime factors pi is a divisor of . ...

References

• Butske, William; Jaje, Lynda M.; Mayernik, Daniel R. (2000). "On the equation , pseudoperfect numbers, and perfectly weighted graphs". Mathematics of Computation 69: 407–420.