In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, a semiperfect number or pseudoperfect 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'' that is equal to the sum of all or some of its
proper 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. 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 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.
...
.
The first few semiperfect numbers are:
6,
12,
18,
20,
24,
28,
30,
36,
40, ...
Properties
* Every
multiple of a semiperfect number is semiperfect.
[Zachariou+Zachariou (1972)] 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 divisible by ...
by any smaller semiperfect number is called ''primitive''.
* Every number of the form 2
''m''''p'' for a natural number ''m'' and 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 ...
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 17 ...
.
* The smallest odd semiperfect number is
945
Year 945 ( CMXLV) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* January 27 – The co-emperors Stephen and Constantine are overthrown barel ...
(see, e.g., Friedman 1993).
* 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
In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 ...
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 two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negat ...
is semiperfect.
* The
natural density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the ...
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.
[Guy (2004) p. 75]
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, ]104 104 may refer to:
*104 (number), a natural number
*AD 104, a year in the 2nd century AD
* 104 BC, a year in the 2nd century BC
* 104 (MBTA bus), Massachusetts Bay Transportation Authority bus route
* Hundred and Four (or Council of 104), a Carthagin ...
, 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
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
:[ there are also infinitely many primitive semiperfect numbers that are not ]harmonic divisor number
In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are:
: 1, 6, 28, ...
s.[
Every semiperfect number is a multiple of a primitive semiperfect number.
]
See also
* Hemiperfect number In number theory, a hemiperfect number is a positive integer with a half-integer abundancy index. In other words, ''σ''(''n'')/''n'' = ''k''/2 for an odd integer ''k'', where ''σ''(''n'') is the divisor function, the sum of all positive divisors ...
* Erdős–Nicolas number
In number theory, an Erdős–Nicolas number is a number that is not perfect, but that equals one of the partial sums of its divisors.
That is, a number is Erdős–Nicolas number when there exists another number such that
: \sum_d=n.
The fir ...
Notes
References
*
* Section B2.
*
*
External links
*
*
{{Classes of natural numbers
Integer sequences
Perfect numbers
de:Vollkommene Zahl#Pseudovollkommene Zahlen