Granville number

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

The Granville set[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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