Granville number

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

Contents

• 
  1 The Granville set
  


• 
  2 General properties
  
  • 
    2.1 S-deficient numbers
    
  
  
  • 
    2.2 S-perfect numbers
    
  
  
  • 
    2.3 S-abundant numbers
    
  
  


• 
  3 Examples
  


• 
  4 References
  

The Granville set

In 1996, Andrew Granville proposed the following construction of the set Sdisplaystyle mathcal S:^[1]

Let 1∈Sdisplaystyle 1in mathcal S and for all n∈N,n>1displaystyle nin mathbb N ,;n>1 let n∈Sdisplaystyle nin mathcal S if:

∑d∣n,d<n,d∈Sd≤ndisplaystyle sum _dmid n,;d<n,;din mathcal Sdleq n

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

General properties

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

S-deficient numbers

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

S-perfect numbers

Numbers that fulfill equality in the above definition are known as Sdisplaystyle mathcal S-perfect numbers.^[1] That is, the Sdisplaystyle mathcal S-perfect numbers are the natural numbers that are equal the sum of their divisors in Sdisplaystyle mathcal S. The first few Sdisplaystyle 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 Sdisplaystyle mathcal S-perfect.^[1] However, there are numbers such as 24 which are Sdisplaystyle mathcal S-perfect but not perfect. The only known Sdisplaystyle 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 Sdisplaystyle mathcal S-abundant numbers. That is, the Sdisplaystyle mathcal S-abundant numbers are the natural numbers that are strictly greater than the sum of their divisors in Sdisplaystyle mathcal S; they belong to the complement of Sdisplaystyle mathcal S. The first few Sdisplaystyle 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 Sdisplaystyle mathcal S because the restriction of the divisors sum to members of Sdisplaystyle mathcal S either decreases the divisors sum or leaves it unchanged. The first natural number that is not in Sdisplaystyle mathcal S is the smallest abundant number, which is 12. The next two abundant numbers, 18 and 20, are also not in Sdisplaystyle mathcal S. However, the fourth abundant number, 24, is in Sdisplaystyle mathcal S because the sum of its proper divisors in Sdisplaystyle mathcal S is:

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

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

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

References

1. 
   ^ ^abcde 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. 
   ^ ^ab de Koninck, J.M. (2009). Those fascinating numbers. AMS Bookstore. p. 40. ISBN 0-8218-4807-0. 
   

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