- Unitary perfect number
A unitary perfect number is an
integer which is the sum of its positive properunitary divisor s, not including the number itself. (Adivisor "d" of a number "n" is a unitary divisor if "d" and "n"/"d" share no common factors.) Someperfect number s 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 OEIS|id=A002827
There are no odd unitary perfect numbers. This follows since one has 2"d"*("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 "p""a" is "p""a" + 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.References
* Section B3.
*
Wikimedia Foundation. 2010.