Primary pseudoperfect number

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Graphical demonstration that 1 = 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/(2×3×11×23×31). Therefore the product, 47058, is primary pseudoperfect.

In mathematics, and particularly in number theory, a primary pseudoperfect number is a number N that satisfies the Egyptian fraction equation

$\sum_{p|N}\frac1p + \frac1N = 1,$

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

$\sum_{p|N}\frac{N}p + 1 = N.$

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, ... (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.

The prime factors of primary pseudoperfect numbers may provide solutions to Znám's problem in which all members of the solution set are prime. For instance, the factors of the primary pseudoperfect number 47058 are the solution set {2,3,11,23,31} to Znám's problem. However, the smaller primary pseudoperfect numbers 2, 6, 42, and 1806 do not correspond to solutions to Znám's problem in this way, as their sets of prime factors violate the requirement in Znám's problem that no number in the set can equal one plus the product of all the other numbers. Anne (1998) observes that there is exactly one solution set of this type that has k primes in it, for each k ≤ 8, and conjectures that the same is true for larger k.

If a primary pseudoperfect number N is one less than a prime number, then N×(N+1) is also primary pseudoperfect. For instance, 47058 is primary pseudoperfect, and 47059 is prime, so 47058 × 47059 = 2214502422 is also primary pseudoperfect.