A unitary perfect number is an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

which is the sum of its positive proper

unitary divisor In mathematics, a natural number ''a'' is a unitary divisor (or Hall divisor) of a number ''b'' if ''a'' is a divisor of ''b'' and if ''a'' and \frac are coprime, having no common factor other than 1. Equivalently, a divisor ''a'' of ''b'' is a un ...

s, not including the number itself. (A

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

''d'' of a number ''n'' is a unitary divisor if ''d'' and ''n''/''d'' share no common factors). The number 6 is the only number that is both a

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...

and a unitary perfect number.

Known examples

The number 60 is a unitary perfect number because 1, 3, 4, 5, 12, 15, and 20 are its proper unitary divisors, and 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60. The first five, and only known, unitary perfect numbers are: *

6 = 2 \times 3

60 = 2^2 \times 3 \times 5

90 = 2 \times 3^2 \times 5

87360 = 2^6 \times 3 \times 5 \times 7 \times 13

, and *

146361946186458562560000 = 2^ \times 3 \times 5^4 \times 7 \times 11 \times 13 \times 19 \times 37 \times 79 \times 109 \times 157 \times 313

. The respective sums of their proper unitary divisors are as follows: * 6 = 1 + 2 + 3 * 60 = 1 + 3 + 4 + 5 + 12 + 15 + 20 * 90 = 1 + 2 + 5 + 9 + 10 + 18 + 45 * 87360 = 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 64 + 65 + 91 + 105 + 192 + 195 + 273 + 320 + 448 + 455 + 832 + 960 + 1344 + 1365 + 2240 + 2496 + 4160 + 5824 + 6720 + 12480 + 17472 + 29120 * 146361946186458562560000 = 1 + 3 + 7 + 11 + ... + 13305631471496232960000 + 20908849455208366080000 + 48787315395486187520000 (4095 divisors in the sum)

Properties

There are no odd unitary perfect numbers. This follows since 2^{''d''*(''n'')} divides the sum of the unitary divisors of an odd number ''n'', where ''d''*(''n'') is the number of distinct

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

factors of ''n''. One gets this because the sum of all the unitary divisors is a

multiplicative function In number theory, a multiplicative function is an arithmetic function f of a positive integer n with the property that f(1)=1 and f(ab) = f(a)f(b) whenever a and b are coprime. An arithmetic function is said to be completely multiplicative (o ...

and one has that the sum of the unitary divisors of a

prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 1 ...

''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 is not known whether or not there are infinitely many unitary perfect numbers, or indeed whether there are any further examples beyond the five already known. A sixth such number would have at least nine distinct odd prime factors.

References

* Section B3. * * * {{Divisor classes Integer sequences Perfect numbers