Granville number
In mathematics, specifically number theory, Granville numbers, also known as [math]\displaystyle{ \mathcal{S} }[/math]-perfect numbers, are an extension of the perfect numbers.
The Granville set
In 1996, Andrew Granville proposed the following construction of a set [math]\displaystyle{ \mathcal{S} }[/math]:[1]
- Let [math]\displaystyle{ 1\in\mathcal{S} }[/math], and for any integer [math]\displaystyle{ n }[/math] larger than 1, let [math]\displaystyle{ n\in{\mathcal{S}} }[/math] if
- [math]\displaystyle{ \sum_{d\mid n, \; d\lt n,\; d\in\mathcal{S}} d \leq n. }[/math]
A Granville number is an element of [math]\displaystyle{ \mathcal{S} }[/math] for which equality holds, that is, [math]\displaystyle{ n }[/math] is a Granville number if it is equal to the sum of its proper divisors that are also in [math]\displaystyle{ \mathcal{S} }[/math]. Granville numbers are also called [math]\displaystyle{ \mathcal{S} }[/math]-perfect numbers.[2]
General properties
The elements of [math]\displaystyle{ \mathcal{S} }[/math] can be k-deficient, k-perfect, or k-abundant. In particular, 2-perfect numbers are a proper subset of [math]\displaystyle{ \mathcal{S} }[/math].[1]
S-deficient numbers
Numbers that fulfill the strict form of the inequality in the above definition are known as [math]\displaystyle{ \mathcal{S} }[/math]-deficient numbers. That is, the [math]\displaystyle{ \mathcal{S} }[/math]-deficient numbers are the natural numbers for which the sum of their divisors in [math]\displaystyle{ \mathcal{S} }[/math] is strictly less than themselves:
- [math]\displaystyle{ \sum_{d\mid{n},\; d\lt n,\; d\in\mathcal{S}}d \lt {n} }[/math]
S-perfect numbers
Numbers that fulfill equality in the above definition are known as [math]\displaystyle{ \mathcal{S} }[/math]-perfect numbers.[1] That is, the [math]\displaystyle{ \mathcal{S} }[/math]-perfect numbers are the natural numbers that are equal the sum of their divisors in [math]\displaystyle{ \mathcal{S} }[/math]. The first few [math]\displaystyle{ \mathcal{S} }[/math]-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 [math]\displaystyle{ \mathcal{S} }[/math]-perfect.[1] However, there are numbers such as 24 which are [math]\displaystyle{ \mathcal{S} }[/math]-perfect but not perfect. The only known [math]\displaystyle{ \mathcal{S} }[/math]-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 [math]\displaystyle{ \mathcal{S} }[/math]-abundant numbers. That is, the [math]\displaystyle{ \mathcal{S} }[/math]-abundant numbers are the natural numbers for which the sum of their divisors in [math]\displaystyle{ \mathcal{S} }[/math] is strictly greater than themselves:
- [math]\displaystyle{ \sum_{d\mid{n},\; d\lt n,\; d\in\mathcal{S}}d \gt {n} }[/math]
They belong to the complement of [math]\displaystyle{ \mathcal{S} }[/math]. The first few [math]\displaystyle{ \mathcal{S} }[/math]-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 [math]\displaystyle{ \mathcal{S} }[/math] because the restriction of the divisors sum to members of [math]\displaystyle{ \mathcal{S} }[/math] either decreases the divisors sum or leaves it unchanged. The first natural number that is not in [math]\displaystyle{ \mathcal{S} }[/math] is the smallest abundant number, which is 12. The next two abundant numbers, 18 and 20, are also not in [math]\displaystyle{ \mathcal{S} }[/math]. However, the fourth abundant number, 24, is in [math]\displaystyle{ \mathcal{S} }[/math] because the sum of its proper divisors in [math]\displaystyle{ \mathcal{S} }[/math] is:
- 1 + 2 + 3 + 4 + 6 + 8 = 24
In other words, 24 is abundant but not [math]\displaystyle{ \mathcal{S} }[/math]-abundant because 12 is not in [math]\displaystyle{ \mathcal{S} }[/math]. In fact, 24 is [math]\displaystyle{ \mathcal{S} }[/math]-perfect - it is the smallest number that is [math]\displaystyle{ \mathcal{S} }[/math]-perfect but not perfect.
The smallest odd abundant number that is in [math]\displaystyle{ \mathcal{S} }[/math] is 2835, and the smallest pair of consecutive numbers that are not in [math]\displaystyle{ \mathcal{S} }[/math] are 5984 and 5985.[1]
References
- ↑ 1.0 1.1 1.2 1.3 1.4 "On a Sum of Divisors Problem". Publications de l'Institut mathématique 64 (78): 9–20. 1996. http://www.emis.de/journals/PIMB/078/n078p009.pdf. Retrieved 27 March 2011.
- ↑ 2.0 2.1 de Koninck, Jean-Marie (2008). Those Fascinating Numbers. Providence, RI: American Mathematical Society. p. 40. ISBN 978-0-8218-4807-4. OCLC 317778112. https://archive.org/details/thosefascinating0000koni/page/40/mode/2up.
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