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 equality ''n'' = 1 + ''k''(''σ''(''n'') − ''n'' − 1) holds, where ''σ''(''n'') 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 divisible by ...
s of ''n''). A hyperperfect number is a ''k''-hyperperfect number for some integer ''k''. Hyperperfect numbers generalize
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, which are 1-hyperperfect.
The first few numbers in the sequence of ''k''-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... , with the corresponding values of ''k'' being 1, 2, 1, 6, 3, 1, 12, ... . The first few ''k''-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... .
List of hyperperfect numbers
The following table lists the first few ''k''-hyperperfect numbers for some values of ''k'', 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 ...
(OEIS) of the sequence of ''k''-hyperperfect numbers:
It can be shown that if ''k'' > 1 is an odd
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
and ''p'' = (3''k'' + 1) / 2 and ''q'' = 3''k'' + 4 are
prime number
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 way ...
s, then ''p''²''q'' is ''k''-hyperperfect; Judson S. McCranie has conjectured in 2000 that all ''k''-hyperperfect numbers for odd ''k'' > 1 are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if ''p'' ≠ ''q'' are odd primes and ''k'' is an integer such that ''k''(''p'' + ''q'') = ''pq'' - 1, then ''pq'' is ''k''-hyperperfect.
It is also possible to show that if ''k'' > 0 and ''p'' = ''k'' + 1 is prime, then for all ''i'' > 1 such that ''q'' = ''p''''i'' − ''p'' + 1 is prime, ''n'' = ''p''''i'' − 1''q'' is ''k''-hyperperfect. The following table lists known values of ''k'' and corresponding values of ''i'' for which ''n'' is ''k''-hyperperfect:
Hyperdeficiency
The newly introduced mathematical concept of hyperdeficiency is related to the hyperperfect numbers.
Definition (Minoli 2010): For any integer ''n'' and for integer ''k'', , define the k-hyperdeficiency (or simply the hyperdeficiency) for the number ''n'' as
δk(n) = n(k+1) +(k-1) – kσ(n)
A number ''n'' is said to be k-hyperdeficient if δ''k''(''n'') > 0.
Note that for ''k''=1 one gets δ1(''n'')= 2''n''–σ(''n''), which is the standard traditional definition of
deficiency
A deficiency is generally a lack of something. It may also refer to:
*A deficient number, in mathematics, a number ''n'' for which ''σ''(''n'') < 2''n''
* MathWorld: Hyperperfect number