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.
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).
Benkoski, S. J. "Elementary Problem and Solution E2308." Amer. Math. Monthly79, 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. Vesnik2, 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.