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.
Famous quotes containing the words perfect and/or number:
“I am not a perfect servant. I am a public servant doing my best against the odds. As I develop and serve, be patient. God is not finished with me yet.”
—Jesse Jackson (b. 1941)
“Envy has blackened every page of his history.... The future, in its justice, will number him among those men whom passions and an excess of activity have condemned to unhappiness, through the gift of genius.”
—Eugène Delacroix (1798–1863)