Primitive semiperfect number
Encyclopedia
In number theory
, 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 odd primitive semiperfect numbers (the smallest is 945), as well as infinitely many primitive semiperfect numbers that are not harmonic divisor number
s.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a semiperfect number
Semiperfect number
In number theory, a semiperfect number or pseudoperfect number is a natural number 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....
that has no semiperfect proper divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
.
The first few primitive semiperfect numbers are 6, 20
20 (number)
20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.-In mathematics:*20 is the basis for vigesimal number systems....
, 28
28 (number)
28 is the natural number following 27 and preceding 29.-In mathematics:It is a composite number, its proper divisors being 1, 2, 4, 7, and 14....
, 88
88 (number)
88 is the natural number following 87 and preceding 89.- In mathematics :Eighty-Eight is a refactorable number, a primitive semiperfect number and an untouchable number...
, 104
104 (number)
104 is the natural number following 103 and preceding 105.- In mathematics :One hundred [and] four is a primitive semiperfect number and a composite number, with its divisors being 1, 2, 4, 8, 13, 26, 52 and 104. As it has 8 divisors total, and 8 is one of those divisors, 104 is a refactorable...
, 272, 304, 350, ...
There are infinitely many odd primitive semiperfect numbers (the smallest is 945), as well as infinitely many primitive semiperfect numbers that are not harmonic divisor number
Harmonic divisor number
In mathematics, a harmonic divisor number, or Ore number , is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are...
s.