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.