Granville number

In mathematics, specifically number theory, Granville numbers are an extension of the perfect numbers.

The Granville set

In 1996, Andrew Granville proposed the following construction of the set $\mathcal{S}$:^[1]

Let $1\in\mathcal{S}$ and for all $n\in{\mathbb{N}},\;n>1$ 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 i.e. 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.^[2]

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 $\mathcal{S}$.^[1]

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 that are strictly less than the sum of their divisors in $\mathcal{S}$.

S-perfect numbers

Numbers that fulfill equality in the above definition are known as $\mathcal{S}$-perfect numbers.^[1] 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 OEIS)

Every perfect number is also $\mathcal{S}$-perfect.^[1] 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 .^[2]

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 that are strictly greater than the sum of their divisors in $\mathcal{S}$; 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 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.^[1]

References

1. De Koninck J-M, Ivić A (1996). "On a Sum of Divisors Problem" (PDF). Publications de l'Institut mathématique 64 (78): 9–20. Retrieved 27 March 2011.
2. de Koninck, J.M. (2009). Those fascinating numbers. AMS Bookstore. p. 40. ISBN 0-8218-4807-0.