Unitary perfect number
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A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself.
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
There are no odd unitary perfect numbers. This follows since one has 2f(n) dividing the sum of the unitary divisors of an odd number. Here f(n) is the number of distinct 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 as pa +1 which is even for all odd primes p. One gets easily from this that 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 even unitary perfect numbers.
See also: perfect number