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 semiperfect number or pseudoperfect 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'' that is equal to the sum of all or some of its
proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a
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 ...
.
The first few semiperfect numbers are:
6,
12,
18,
20,
24,
28,
30,
36,
40, ...
Properties
* Every
multiple of a semiperfect number is semiperfect. A semiperfect number that is not
divisible
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 ...
by any smaller semiperfect number is called ''primitive''.
* Every number of the form 2
''m''''p'' for a natural number ''m'' and an
odd 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 ...
''p'' such that ''p'' < 2
''m''+1 is also semiperfect.
** In particular, every number of the form 2
''m''(2
''m''+1 − 1) is semiperfect, and indeed perfect if 2
''m''+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 1 ...
.
* The smallest odd semiperfect number is
945.
* A semiperfect number is necessarily either perfect or
abundant. An abundant number that is not semiperfect is called a
weird number.
* With the exception of 2, all
primary pseudoperfect numbers are semiperfect.
* Every
practical number that is not a
power of two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number 2, two as the Base (exponentiation), base and integer as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^ ...
is semiperfect.
* The
natural density of the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of semiperfect numbers exists.
Primitive semiperfect numbers
A primitive semiperfect number (also called a ''primitive pseudoperfect number'', ''irreducible semiperfect number'' or ''irreducible pseudoperfect number'') is a semiperfect number that has no semiperfect proper divisor.
The first few primitive semiperfect numbers are
6,
20,
28,
88,
104,
272,
304,
350, ...
There are infinitely many such numbers. All numbers of the form 2
''m''''p'', with ''p'' a prime between 2
''m'' and 2
''m''+1, are primitive semiperfect, but this is not the only form: for example, 770. There are infinitely many odd primitive semiperfect numbers, the smallest being 945, a result of
Paul Erdős. There are also infinitely many primitive semiperfect numbers that are not
harmonic divisor numbers.
Every semiperfect number is a multiple of a primitive semiperfect number.
See also
*
Hemiperfect number
*
Erdős–Nicolas number
Notes
References
*
* Section B2.
*
*
External links
*
*
{{Classes of natural numbers
Integer sequences
Perfect numbers
de:Vollkommene Zahl#Pseudovollkommene Zahlen