In mathematics, a superperfect number is a positive
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 ...
''n'' that satisfies
:
where σ is the
divisor summatory function
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 ...
. Superperfect numbers are a generalization of
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. The term was coined by D. Suryanarayana (1969).
The first few superperfect numbers are :
:
2,
4,
16,
64,
4096,
65536
65536 is the natural number following 65535 and preceding 65537.
65536 is a power of two: 2^ (2 to the 16th power).
65536 is the smallest number with ''exactly'' 17 divisors.
In mathematics
65536 is 2^, so in tetration notation 65536 is ...
, 262144, 1073741824, ... .
To illustrate: it can be seen that 16 is a superperfect number as σ(16) = 1 + 2 + 4 + 8 + 16 = 31, and σ(31) = 1 + 31 = 32, thus σ(σ(16)) = 32 = 2 × 16.
If ''n'' is an ''even'' superperfect number, then ''n'' must be a power of 2, 2
''k'', such that 2
''k''+1 − 1 is a
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
.
It is not known whether there are any
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 ...
superperfect numbers. An odd superperfect number ''n'' would have to be a square number such that either ''n'' or σ(''n'') is divisible by at least three distinct primes.
There are no odd superperfect numbers below 7.
[Guy (2004) p. 99.]
Generalizations
Perfect and superperfect numbers are examples of the wider class of ''m''-superperfect numbers, which satisfy
:
corresponding to ''m''=1 and 2 respectively. For ''m'' ≥ 3 there are no even ''m''-superperfect numbers.
The ''m''-superperfect numbers are in turn examples of (''m'',''k'')-perfect numbers which satisfy
:
With this notation, perfect numbers are (1,2)-perfect,
multiperfect numbers are (1,''k'')-perfect, superperfect numbers are (2,2)-perfect and ''m''-superperfect numbers are (''m'',2)-perfect.
[Guy (2007) p.79] Examples of classes of (''m'',''k'')-perfect numbers are:
:
Notes
References
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*
*
*
*
{{Classes of natural numbers
Divisor function
Integer sequences
Unsolved problems in number theory