Semiperfect number
From Wikipedia, the free encyclopedia
| Divisibility-based sets of integers |
| Form of factorization: |
| Prime number |
| Composite number |
| Powerful number |
| Square-free number |
| Achilles number |
| Constrained divisor sums: |
| Perfect number |
| Almost perfect number |
| Quasiperfect number |
| Multiply perfect number |
| Hyperperfect number |
| Unitary perfect number |
| Semiperfect number |
| Primitive semiperfect number |
| Practical number |
| Numbers with many divisors: |
| Abundant number |
| Highly abundant number |
| Superabundant number |
| Colossally abundant number |
| Highly composite number |
| Superior highly composite number |
| Other: |
| Deficient number |
| Weird number |
| Amicable number |
| Friendly number |
| Sociable number |
| Solitary number |
| Sublime number |
| Harmonic divisor number |
| Frugal number |
| Equidigital number |
| Extravagant number |
| See also: |
| Divisor function |
| Divisor |
| Prime factor |
| Factorization |
In mathematics, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors.
The first few semiperfect numbers are
every multiple of a semiperfect number is semiperfect, and every number of the form 2mp for a natural number m and a prime number p such that p < 2m + 1 is also semiperfect.
The smallest odd semiperfect number is 945 (see, e.g., Friedman 1993).
A semiperfect number that is equal to the sum of all its proper divisors is called a perfect number; an abundant number which is not semiperfect is called a weird number. With one exception, all primary pseudoperfect numbers are semiperfect. Every practical number that is not a power of two is semiperfect.
A semiperfect number that is not divisible by any smaller semiperfect number is a primitive semiperfect number.
[edit] References
- Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". Journal of Number Theory 44: 328–339. doi:10.1006/jnth.1993.1057. MR1233293.
