Primary pseudoperfect number

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, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation

$\sum _{{p|N}}{\frac 1p}+{\frac 1N}=1,$

where the sum is over only the prime divisors of N.

Properties

Equivalently, N is a primary pseudoperfect number if it satisfies

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

Except for the primary pseudoperfect number N = 2, this expression gives a representation for N as the sum of distinct divisors of N. Therefore, each primary pseudoperfect number N (except N = 2) is also pseudoperfect.

The first few primary pseudoperfect numbers are

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

The first four of these numbers are one less than the corresponding numbers in Sylvester's sequence, but then the two sequences diverge.

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 sometimes may provide solutions to Znám's problem, in which all elements of the solution set are prime. For instance, the prime factors of the primary pseudoperfect number 47058 form 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 that no number in the set can equal one plus the product of 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.

History

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

See also

• Giuga number

References

• Anne, Premchand (1998), "Egyptian fractions and the inheritance problem", The College Mathematics Journal, The College Mathematics Journal, Vol. 29, No. 4, 29 (4): 296–300, doi:10.2307/2687685, JSTOR 2687685.

• Butske, William; Jaje, Lynda M.; Mayernik, Daniel R. (2000), "On the equation $\scriptstyle \sum _{{p|N}}{\frac {1}{p}}+{\frac {1}{N}}=1$, pseudoperfect numbers, and perfectly weighted graphs", Mathematics of Computation, 69: 407–420, doi:10.1090/S0025-5718-99-01088-1.

External links

• Primary Pseudoperfect Number at PlanetMath.org.
• Weisstein, Eric W. "Primary Pseudoperfect Number". MathWorld.