unitary perfect number

A unitary perfect number is an integer which is the sum of its positive proper unitary divisorsMathworldPlanetmath, not including the number itself. (A divisorMathworldPlanetmathPlanetmath d of a number n is a unitary divisor if d and nd share no common factors.) Some perfect numbers are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers.

The first few unitary perfect numbers are 6, 60, 90, 87360, 146361946186458562560000 (sequence A002827 in the 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 numberMathworldPlanetmathPlanetmath (where d*(n) is the number of distinct prime divisorsPlanetmathPlanetmath of n). One gets this because the sum of all the unitary divisors is a multiplicative functionMathworldPlanetmath 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.

Title unitary perfect number
Canonical name UnitaryPerfectNumber
Date of creation 2013-03-22 16:19:29
Last modified on 2013-03-22 16:19:29
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 4
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A05