| 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 |
| Superperfect 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 primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a semiperfect natural number that has no semiperfect proper divisor.
The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, ... (sequence A006036 in OEIS).
There are infinitely many odd primitive semiperfect numbers (the smallest is 945,) as well as infinitely many primitive semiperfect numbers that are not harmonic divisor numbers.
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