In
mathematics a primitive abundant number is an
abundant number
In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. Th ...
whose
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 are all
deficient number
In number theory, a deficient number or defective number is a number ''n'' for which the sum of divisors of ''n'' is less than 2''n''. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than ''n''. For ex ...
s.
For example, 20 is a primitive abundant number because:
:#The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number.
:#The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number.
The first few primitive abundant numbers are:
:
20,
70,
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, 368, 464, 550, 572 ...
The smallest odd primitive abundant number is 945.
A variant definition is abundant numbers having no abundant proper divisor . It starts:
:
12,
18,
20,
30, 42, 56, 66, 70, 78, 88, 102, 104, 114
Properties
Every multiple of a primitive abundant number is an abundant number.
Every abundant number is a multiple of a primitive abundant number or a multiple of a perfect number.
Every primitive abundant number is either a
primitive semiperfect number or a
weird number
In number theory, a weird number is a natural number that is abundant but not semiperfect.
In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divis ...
.
There are an infinite number of primitive abundant numbers.
The number of primitive abundant numbers less than or equal to ''n'' is
[Paul Erdős, ''Journal of the London Mathematical Society'' 9 (1934) 278–282.]
References
{{Classes of natural numbers
Divisor function
Integer sequences