In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, a -hyperperfect 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 ...
for which the equality
holds, where is 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 ...
(i.e.,
the sum of all positive
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 ). A hyperperfect number is a -hyperperfect number for some integer . Hyperperfect numbers generalize
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...
s, which are 1-hyperperfect.
The first few numbers in the sequence of -hyperperfect numbers are , with the corresponding values of being . The first few -hyperperfect numbers that are not perfect are .
List of hyperperfect numbers
The following table lists the first few -hyperperfect numbers for some values of , together with the sequence number in the
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...
(OEIS) of the sequence of -hyperperfect numbers:
It can be shown that if is an
odd 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 ...
and
and
are
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
s, then is -hyperperfect; Judson S. McCranie has conjectured in 2000 that all -hyperperfect numbers for odd are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if are odd primes and is an integer such that
then is -hyperperfect.
It is also possible to show that if and
is prime, then for all such that
is prime,
is -hyperperfect. The following table lists known values of and corresponding values of for which is -hyperperfect:
References
*
Further reading
Articles
* .
* .
* .
* .
* .
* .
* .
* .
* .
Books
* Daniel Minoli, ''Voice over MPLS'', McGraw-Hill, New York, NY, 2002, (p. 114-134)
External links
MathWorld: Hyperperfect numberA long list of hyperperfect numbers under Data
{{Classes of natural numbers
Divisor function
Integer sequences
Perfect numbers