Unitary perfect number

Unitary perfect number

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 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.
*


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