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In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
''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 divisibl ...
s of ''n'' (the sum-of-divisors function ''σ''(''n'')) is equal to 2''n'' − 1, the sum of all proper 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 ...
s 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 the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hi ...
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 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 ...
''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. Moreover if ''a'' and ''b'' are positive odd integers such that