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 ...

, 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'' (includin ...

''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. Th ...

s of minimal abundance (which is 1).

Theorems

If a quasiperfect number exists, it must be an

odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

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 10³⁵ 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 ...

Numbers do exist where the sum of all the

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. ...

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 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-negati ...

Betrothed numbers Betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors of either number is one more than the value of the other number. In other words, (''m'', ''n'') are a pair of betrothed numbers if ...

relate to quasiperfect numbers like

amicable numbers Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(''a'')=''b'' and σ(''b'')=''a'', where σ(''n'') is equal to the sum of positive di ...

relate to perfect numbers.

Notes

References

* * * * * * Arithmetic dynamics Divisor function Integer sequences Unsolved problems in mathematics {{numtheory-stub

Theorems

Related

Notes

References