Granville number

In mathematics, specifically number theory, Granville numbers, also known as $𝒮$-perfect numbers, are an extension of the perfect numbers.

The Granville set

In 1996, Andrew Granville proposed the following construction of a set $𝒮$:

Let $1\in 𝒮$, and for any integer $n$ larger than 1, let $n\in 𝒮$ if

$\sum _{d\mid n,d<n,d\in 𝒮}d\le n.$

A Granville number is an element of $𝒮$ for which equality holds, that is, $n$ is a Granville number if it is equal to the sum of its proper divisors that are also in $𝒮$. Granville numbers are also called $𝒮$-perfect numbers.

General properties

The elements of $𝒮$ can be Template:Math-deficient, Template:Math-perfect, or Template:Math-abundant. In particular, 2-perfect numbers are a proper subset of $𝒮$.

S-deficient numbers

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

$\sum _{d\mid n,d<n,d\in 𝒮}d<n$

S-perfect numbers

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

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

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

Every number of form 2^(n - 1) * (2^n - 1) * (2^n)^m where m >= 0, where 2^n - 1 is Prime, are Granville Numbers. So, there are infinitely many Granville Numbers and the infinite family has 2 prime factors- 2 and a Mersenne Prime. Others include 126, 5540590, 9078520, 22528935, 56918394 and 246650552 having 3, 5, 5, 5, 5 and 5 prime factors.

S-abundant numbers

Numbers that violate the inequality in the above definition are known as $𝒮$-abundant numbers. That is, the $𝒮$-abundant numbers are the natural numbers for which the sum of their divisors in $𝒮$ is strictly greater than themselves:

$\sum _{d\mid n,d<n,d\in 𝒮}d>n$

They belong to the complement of $𝒮$. The first few $𝒮$-abundant numbers are:

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

Examples

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

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

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

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