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, ... (sequence A118372 in the OEIS)

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.

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:

$\sum _{d\mid {n},\;d<n,\;d\in {\mathcal {S}}}d>{n}$

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, ... (sequence A181487 in the OEIS)

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.