Granville number

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

The Granville set
In 1996, Andrew Granville proposed the following construction of a set $$\mathcal{S}$$:


 * Let $$1\in\mathcal{S}$$, and for any integer $$n$$ larger than 1, let $$n\in{\mathcal{S}}$$ if
 * $$\sum_{d\mid n, \; d<n,\; d\in\mathcal{S}} d \leq n.$$

A Granville number is an element of $$\mathcal{S}$$ 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 $$\mathcal{S}$$. Granville numbers are also called $$\mathcal{S}$$-perfect numbers.

General properties
The elements of $$\mathcal{S}$$ can be $k$-deficient, $k$-perfect, or $k$-abundant. In particular, 2-perfect numbers are a proper subset of $$\mathcal{S}$$.

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


 * $$\sum_{d\mid{n},\; d<n,\; d\in\mathcal{S}}d < {n}$$

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


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

Every perfect number is also $$\mathcal{S}$$-perfect. However, there are numbers such as 24 which are $$\mathcal{S}$$-perfect but not perfect. The only known $$\mathcal{S}$$-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 $$\mathcal{S}$$-abundant numbers. That is, the $$\mathcal{S}$$-abundant numbers are the natural numbers for which the sum of their divisors in $$\mathcal{S}$$ is strictly greater than themselves:


 * $$\sum_{d\mid{n},\; d {n}$$

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


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

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


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

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

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