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In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an almost perfect number (sometimes also called slightly defective or least deficient 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 ...
''n'' such that the sum of all 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 of ''n'' (the sum-of-divisors 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'' (including ...
''σ''(''n'')) is equal to 2''n'' − 1, the sum of all proper divisors of ''n'', ''s''(''n'') = ''σ''(''n'') − ''n'', then being equal to ''n'' − 1. The only known almost perfect numbers are powers of 2
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negative ...
with non-negative exponents . Therefore the only known odd almost perfect number is 20 = 1, and the only known even almost perfect numbers are those of the form 2''k'' for some positive number ''k''; however, it has not been shown that all almost perfect numbers are of this form. It is known that an odd almost perfect number greater than 1 would have at least six 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.
If ''m'' is an odd almost perfect number then is a Descartes number In number theory, a Descartes number is an odd number which would have been an odd perfect number, if one of its composite factors were prime. They are named after René Descartes who observed that the number would be an odd perfect number if on ...
. Moreover if ''a'' and ''b'' are positive odd integers such that