In 1996, Andrew Granville proposed the following construction of the set S :
Let
1 ∈ S and for all
n ∈ N , n > 1 let
n ∈ S if:
∑ d ∣ n , d < n , d ∈ S d ≤ n A Granville number is an element of S for which equality holds i.e. it is equal to the sum of its proper divisors that are also in S . Granville numbers are also called S -perfect numbers.
The elements of S can be k-deficient, k-perfect, or k-abundant. In particular, 2-perfect numbers are a proper subset of S .
Numbers that fulfill the strict form of the inequality in the above definition are known as S -deficient numbers. That is, the S -deficient numbers are the natural numbers that are strictly less than the sum of their divisors in S .
Numbers that fulfill equality in the above definition are known as S -perfect numbers. That is, the S -perfect numbers are the natural numbers that are equal the sum of their divisors in S . The first few 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 -perfect. However, there are numbers such as 24 which are S -perfect but not perfect. The only known S -perfect number with three distinct prime factors is 126 = 2 · 32 · 7 .
Numbers that violate the inequality in the above definition are known as S -abundant numbers. That is, the S -abundant numbers are the natural numbers that are strictly greater than the sum of their divisors in S ; they belong to the complement of S . The first few 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)
Every deficient number and every perfect number is in S because the restriction of the divisors sum to members of S either decreases the divisors sum or leaves it unchanged. The first natural number that is not in S is the smallest abundant number, which is 12. The next two abundant numbers, 18 and 20, are also not in S . However, the fourth abundant number, 24, is in S because the sum of its proper divisors in S is:
1 + 2 + 3 + 4 + 6 + 8 = 24
In other words, 24 is abundant but not S -abundant because 12 is not in S . In fact, 24 is S -perfect - it is the smallest number that is S -perfect but not perfect.
The smallest odd abundant number that is in S is 2835, and the smallest pair of consecutive numbers that are not in S are 5984 and 5985.
