Pseudoperfect Number

A pseudoperfect number, sometimes also called a semiperfect number (Benkoski 1972, Butske et al. 1999), is a positive integer such as  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  is pseudoperfect since

and , , and  are all proper divisors of . Every multiple of a pseudoperfect number is pseudoperfect, as are all numbers  for  and  a prime between  and  (Guy 1994, p. 47).

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

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 , 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.