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The Granville setEdit

In 1996, Andrew Granville proposed the following construction of the set  :[1]

Let   and for all   let   if:
 

A Granville number is an element of   for which equality holds i.e. it is equal to the sum of its proper divisors that are also in  . Granville numbers are also called  -perfect numbers.[2]

General propertiesEdit

The elements of   can be k-deficient, k-perfect, or k-abundant. In particular, 2-perfect numbers are a proper subset of  .[1]

S-deficient numbersEdit

Numbers that fulfill the strict form of the inequality in the above definition are known as  -deficient numbers. That is, the  -deficient numbers are the natural numbers for that the sum of their divisors in   is strictly less than themselves:

 

S-perfect numbersEdit

Numbers that fulfill equality in the above definition are known as  -perfect numbers.[1] That is, the  -perfect numbers are the natural numbers that are equal the sum of their divisors in  . The first few  -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  -perfect.[1] However, there are numbers such as 24 which are  -perfect but not perfect. The only known  -perfect number with three distinct prime factors is 126 = 2 · 32 · 7 .[2]

S-abundant numbersEdit

Numbers that violate the inequality in the above definition are known as  -abundant numbers. That is, the  -abundant numbers are the natural numbers for which the sum of their divisors in   is strictly greater than themselves:

 

They belong to the complement of  . The first few  -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)

ExamplesEdit

Every deficient number and every perfect number is in   because the restriction of the divisors sum to members of   either decreases the divisors sum or leaves it unchanged. The first natural number that is not in   is the smallest abundant number, which is 12. The next two abundant numbers, 18 and 20, are also not in  . However, the fourth abundant number, 24, is in   because the sum of its proper divisors in   is:

1 + 2 + 3 + 4 + 6 + 8 = 24

In other words, 24 is abundant but not  -abundant because 12 is not in  . In fact, 24 is  -perfect - it is the smallest number that is  -perfect but not perfect.

The smallest odd abundant number that is in   is 2835, and the smallest pair of consecutive numbers that are not in   are 5984 and 5985.[1]

ReferencesEdit

  1. ^ a b c d e 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.CS1 maint: uses authors parameter (link)
  2. ^ a b de Koninck, J.M. (2009). Those fascinating numbers. AMS Bookstore. p. 40. ISBN 0-8218-4807-0.