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Pseudoperfect Number


A pseudoperfect number, sometimes also called a semiperfect number (Benkoski 1972, Butske et al. 1999), is a positive integer such as 20=1+4+5+10 which is the sum of some (or all) of its proper divisors. Identifying pseudoperfect numbers is therefore equivalent to solving the subset sum problem.

A pseudoperfect number which is the sum of all its proper divisors is called a perfect number.

The first few pseudoperfect numbers are 6, 12, 18, 20, 24, 28, 30, 36, 40, ... (OEIS A005835).

Every positive integer n=6k is pseudoperfect since

 6k=k+2k+3k

and k, 2k, and 3k are all proper divisors of 6k. Every multiple of a pseudoperfect number is pseudoperfect, as are all numbers 2^mp for m>=1 and p a prime between 2^m and 2^(m+1) (Guy 1994, p. 47).

A pseudoperfect number cannot be deficient (or therefore prime). Rare abundant numbers which are not pseudoperfect are called weird numbers.


See also

Abundant Number, Deficient Number, Perfect Number, Primary Pseudoperfect Number, Primitive Pseudoperfect Number, Subset Sum Problem, Weird Number

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References

Benkoski, S. J. "Elementary Problem and Solution E2308." Amer. Math. Monthly 79, 774, 1972.Benkoski, S. J. and Erdős, P. "On Weird and Pseudoperfect Numbers." Math. Comput. 28, 617-623, 1974.Butske, W.; Jaje, L. M.; and Mayernik, D. R. "The Equation sum_(p|N)1/p+1/N=1, Pseudoperfect Numbers, and Partially Weighted Graphs." Math. Comput. 69, 407-420, 1999.de Koninck, J.-M. Entry 70 in Ces nombres qui nous fascinent. Paris: Ellipses, p. 24, Paris 2008.Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.Hindin, J. "Quasipractical Numbers." IEEE Comm. Mag., 41-45, March 1980.Sierpiński, W. "Sur les numbers psuedoparfaits." Mat. Vesnik 2, 212-213, 1965.Sloane, N. J. A. Sequence A005835/M4094 in "The On-Line Encyclopedia of Integer Sequences."Zachariou, A. and Zachariou, E. "Perfect, Semi-Perfect and Ore Numbers." Bull. Soc. Math. Gréce (New Ser.) 13, 12-22, 1972.

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Pseudoperfect Number

Cite this as:

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

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