Unitary perfect number
Encyclopedia
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 number
s 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
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.
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
which is the sum of its positive proper unitary divisors, not including the number itself. (A divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
d of a number n is a unitary divisor if d and n/d share no common factors.) Some perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
s are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers.
Thus, 60
60 (number)
60 is the natural number following 59 and preceding 61. Being three times twenty, 60 is called "three score" in some older literature.-In mathematics:...
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
90 (number)
90 is the natural number preceded by 89 and followed by 91.- In mathematics :*Because 90 is the sum of its unitary divisors , it is a unitary perfect number, and because it is equal to the sum of a subset of its divisors, it is a semiperfect number. 90 is a pronic number. But it is also a...
, 87360, 146361946186458562560000
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
Multiplicative function
In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
and one has the sum of the unitary divisors of a power of a prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
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.