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In mathematics, and in particular number theory, a primary pseudoperfect number is a number N that satisfies the Egyptian fraction equation: 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. 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, ... . 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. See also Giuga number.
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