Sources
Help build the largest human-edited directory on the web.
Submit a site - Become an editor
Wikipedia. Licensed under the GNU Free Documentation License.
Are you an expert in this subject? Join the discussion and share your knowledge at Wikipedia.org.
Unitary perfect 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 |
| Sociable number |
| Sublime number |
| Harmonic divisor number |
| Frugal number |
| Equidigital number |
| Extravagant number |
| See also: |
| Divisor function |
| Divisor |
| Prime factor |
| Factorization |
A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. (A divisor d of a number n is a unitary divisor if d and n/d share no common factors.) Some perfect numbers are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers.
Thus, 60 is a unitary perfect number, because its unitary divisors, 1, 3, 4, 5, 12, 15 and 20 are its proper unitary divisors, and 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60. The first few unitary perfect numbers are:
6, 60, 90, 87360, 146361946186458562560000 (sequence A002827 in OEIS)
There are no odd unitary perfect numbers. This follows since one has 2d*(n) dividing the sum of the unitary divisors of an odd number (where d*(n) is the number of distinct prime divisors of n). One gets this because the sum of all the unitary divisors is a multiplicative function and one has the sum of the unitary divisors of a power of a prime pa is pa + 1 which is even for all odd primes p. Therefore, an odd unitary perfect number must have only one distinct prime factor, and it is not hard to show that a power of prime cannot be a unitary perfect number, since there are not enough divisors. It's not known whether or not there are infinitely many unitary perfect numbers.
 |
This number theory-related article is a stub. You can help Wikipedia by expanding it. |