In
mathematics, a quasiperfect number is a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
''n'' for which the sum of all its
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 divisible by ...
s (the
divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includi ...
''σ''(''n'')) is equal to 2''n'' + 1. Equivalently, ''n'' is the sum of its non-trivial divisors (that is, its divisors excluding 1 and ''n''). No quasiperfect numbers have been found so far.
The quasiperfect numbers are the
abundant number
In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The ...
s of minimal abundance (which is 1).
Theorems
If a quasiperfect number exists, it must be an
odd square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The u ...
greater than 10
35 and have at least seven distinct
prime factor
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 ...
s.
Related
Numbers do exist where the sum of all the
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 divisible by ...
s ''σ''(''n'') is equal to 2''n'' + 2: 20, 104, 464, 650, 1952, 130304, 522752 ... .
Many of these numbers are of the form 2
''n''−1(2
''n'' − 3) where 2
''n'' − 3 is prime (instead of 2
''n'' − 1 with
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
s). In addition,
numbers exist where the sum of all the divisors ''σ''(''n'') is equal to 2''n'' − 1, such as the
powers of 2.
Betrothed numbers relate to quasiperfect numbers like
amicable numbers relate to perfect numbers.
Notes
References
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Arithmetic dynamics
Divisor function
Integer sequences
Unsolved problems in mathematics
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