In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically number theory

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...

, Granville numbers are an extension of the perfect number

Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

s.

The Granville set

In 1996, Andrew Granville

Andrew Granville

Andrew James Granville is a British mathematician, working in the field of number theory.He has been a faculty member at the Université de Montréal since 2002. Before moving to Montreal he was a mathematics professor at University of Georgia from 1991 until 2002...

 proposed the following construction of the set

Set

A set is a collection of well defined and distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from...

 :

Let and for all let if:

A Granville number is an element of for which strict equality holds i.e. it is equal to the sum of its proper divisors that are also in . Granville numbers are also called -perfect numbers.

General properties

The elements of can be -deficient, -perfect, or -abundant. In particular, 2-perfect numbers

Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

 are a proper subset of .

S-deficient numbers

Numbers that fulfill the strict form of the inequality in the above definition are known as -deficient numbers. That is, the -deficient numbers are the natural numbers that are strictly less than the sum of their divisors in .

S-perfect numbers

Numbers that fulfill equality in the above definition are known as -perfect numbers. That is, the -perfect numbers are the natural numbers that are equal the sum of their divisors in . The first few -perfect numbers are:

6, 24, 28, 96, 126, 224, 384, 496, 1536, 1792, 6144, 8128, 14336, ...

Every perfect number

Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

 is also -perfect. However, there are numbers such as 24 which are -perfect but not perfect. The only known -perfect number with three distinct prime factors is 126 = 2 · 3² · 7 .

S-abundant numbers

Numbers that violate the inequality in the above definition are known as -abundant numbers. That is, the -abundant numbers are the natural numbers that are strictly greater than the sum of their divisors in ; they belong to the complement

Complement (set theory)

In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

 of . The first few -abundant numbers are:

12, 18, 20, 30, 42, 48, 56, 66, 70, 72, 78, 80, 84, 88, 90, 102, 104, ...

Examples

Every deficient number

Deficient number

In number theory, a deficient number or defective number is a number n for which the sum of divisors σIn number theory, a deficient number or defective number is a number n for which the sum of divisors σIn number theory, a deficient number or defective number is a number n for which...

 and every perfect number

Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

 is in because the restriction of the divisors sum to members of either decreases the divisors sum or leaves it unchanged. The first natural number that is not in is the smallest abundant number, which is 12. The next two abundant numbers, 18 and 20, are also not in . However, the fourth abundant number, 24, is in because the sum of its proper divisors in is:

1 + 2 + 3 + 4 + 6 + 8 = 24

In other words, 24 is abundant but not -abundant because 12 is not in . In fact, 24 is -perfect - it is the smallest number that is -perfect but not perfect.

The smallest odd abundant number that is in is 2835, and the smallest pair of consecutive numbers that are not in are 5984 and 5985.