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 k-deficient, k-perfect, or k-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, ... (sequence A118372 in the 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.

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, ... (sequence A181487 in the 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.