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.
, 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