Semiperfect

Semiperfect numbers, also called pseudoperfect numbers, are the positive integers n for which some subset of the proper divisors of n sums to n. Proper divisors are all positive divisors of n excluding n itself. Equivalently, there exists a subset D of the divisors d of n with d < n such that the sum of D equals n.

Common examples illustrate the idea. The number 6 is semiperfect because 6 = 1 + 2 + 3. The number 12 is semiperfect since 12 = 2 + 4 + 6. In fact, many small highly composite or abundant numbers are semiperfect; 18 = 9 + 6 + 3 and 20 = 10 + 5 + 4 + 1 also work. If a number is perfect, it is automatically semiperfect, since the sum of all proper divisors equals the number.

There are abundant numbers that are not semiperfect, known as weird numbers. The smallest example is 70; it has a sum of proper divisors equal to 74, but no subset of those divisors adds up to 70. This shows that not all abundant numbers are semiperfect.

Relations to other concepts include that every practical number is semiperfect, and semiperfect numbers form a broad class within the non-deficient integers (those whose proper divisors sum to at least the number). Determining semiperfectness is related to the subset sum problem and generally lacks a simple closed formula. Semiperfect numbers are of interest in number theory for their structural and additive properties.