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Primary Pseudoperfect Number

A primary pseudoperfect number is a positive integer N>1 that is a product of distinct prime numbers and satisfies

1/N+sum_(p|N)1/p=1,

where the sum is over the prime number divisors of N (Butske et al. 2000). The first few are 2, 6, 42, 1806, 47058, ... (OEIS A054377).

The similar equation

-1/N+sum_(p|N)1/p=1

arises in the definition of Giuga numbers.

See also

Giuga Number, Pseudoperfect Number

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References

Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgensohn, R. "Giuga's Conjecture on Primality." Amer. Math. Monthly 103, 40-50, 1996.Butske, W.; Jaje, L. M.; and Mayernik, D. R. "On the Equation sum_(p|N)1/p+1/N=1, Pseudoperfect Numbers, and Perfectly Weighted Graphs." Math. Comput. 69, 407-420, 2000. https://doi.org/10.1090/S0025-5718-99-01088-1.Cao, Z.; Liu, R.; and Zhang, L. "On the Equation sum_(j=1)^(s)(1/x_j)+1/(x_1...x_s)=1 and Znám's Problem." J. Number Th. 27, 206-211, 1987. https://doi.org/10.1016/0022-314X(87)90062-X.Ke, Z. and Sun, Q. "On the Representation of 1 by Unit Fractions." Sichuan Daxue Xuebao 1, 13-29, 1964.Sloane, N. J. A. Sequence A054377 in "The On-Line Encyclopedia of Integer Sequences."

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Primary Pseudoperfect Number

Cite this as:

Weisstein, Eric W. "Primary Pseudoperfect Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PrimaryPseudoperfectNumber.html

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