Granville number

In mathematics, specifically number theory, Granville numbers, also known as S {\displaystyle {\mathcal {S}}} -perfect numbers, are an extension of the perfect numbers.

The Granville set

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

Let 1 S {\displaystyle 1\in {\mathcal {S}}} , and for any integer n {\displaystyle n} larger than 1, let n S {\displaystyle n\in {\mathcal {S}}} if
d n , d < n , d S d n . {\displaystyle \sum _{d\mid n,\;d<n,\;d\in {\mathcal {S}}}d\leq n.}

A Granville number is an element of S {\displaystyle {\mathcal {S}}} for which equality holds, that is, n {\displaystyle n} is a Granville number if it is equal to the sum of its proper divisors that are also in S {\displaystyle {\mathcal {S}}} . Granville numbers are also called S {\displaystyle {\mathcal {S}}} -perfect numbers.[2]

General properties

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

S-deficient numbers

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

d n , d < n , d S d < n {\displaystyle \sum _{d\mid {n},\;d<n,\;d\in {\mathcal {S}}}d<{n}}

S-perfect numbers

Numbers that fulfill equality in the above definition are known as S {\displaystyle {\mathcal {S}}} -perfect numbers.[1] That is, the S {\displaystyle {\mathcal {S}}} -perfect numbers are the natural numbers that are equal the sum of their divisors in S {\displaystyle {\mathcal {S}}} . The first few S {\displaystyle {\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 S {\displaystyle {\mathcal {S}}} -perfect.[1] However, there are numbers such as 24 which are S {\displaystyle {\mathcal {S}}} -perfect but not perfect. The only known S {\displaystyle {\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 S {\displaystyle {\mathcal {S}}} -abundant numbers. That is, the S {\displaystyle {\mathcal {S}}} -abundant numbers are the natural numbers for which the sum of their divisors in S {\displaystyle {\mathcal {S}}} is strictly greater than themselves:

d n , d < n , d S d > n {\displaystyle \sum _{d\mid {n},\;d<n,\;d\in {\mathcal {S}}}d>{n}}

They belong to the complement of S {\displaystyle {\mathcal {S}}} . The first few S {\displaystyle {\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 S {\displaystyle {\mathcal {S}}} because the restriction of the divisors sum to members of S {\displaystyle {\mathcal {S}}} either decreases the divisors sum or leaves it unchanged. The first natural number that is not in S {\displaystyle {\mathcal {S}}} is the smallest abundant number, which is 12. The next two abundant numbers, 18 and 20, are also not in S {\displaystyle {\mathcal {S}}} . However, the fourth abundant number, 24, is in S {\displaystyle {\mathcal {S}}} because the sum of its proper divisors in S {\displaystyle {\mathcal {S}}} is:

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

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

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

References

  1. ^ a b c d e De Koninck JM, Ivić A (1996). "On a Sum of Divisors Problem" (PDF). Publications de l'Institut mathématique. 64 (78): 9–20. Retrieved 27 March 2011.
  2. ^ a b de Koninck, Jean-Marie (2008). Those Fascinating Numbers. Translated by de Koninck, J. M. Providence, RI: American Mathematical Society. p. 40. ISBN 978-0-8218-4807-4. MR 2532459. OCLC 317778112.