Granville number explained
In mathematics, specifically number theory, Granville numbers, also known as
-perfect numbers, are an extension of the
perfect numbers.
The Granville set
:
[1] Let
, and for any integer
larger than 1, let
if
} d \leq n.
A Granville number is an element of
for which equality holds, that is,
is a Granville number if it is equal to the sum of its proper divisors that are also in
. Granville numbers are also called
-perfect numbers.
[2] General properties
The elements of
can be -deficient, -perfect, or -abundant. In particular,
2-perfect numbers are a proper subset of
.
S-deficient numbers
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 which the sum of their divisors in
is strictly less than themselves:
\sumd\mid{n, d<n, d\inl{S}}d<{n}
S-perfect numbers
Numbers that fulfill equality in the above definition are known as
-perfect numbers. 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, ...
Every perfect number is also
-perfect. 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 · 3
2 · 7.
Every number of form 2^(n - 1) * (2^n - 1) * (2^n)^m where m >= 0, where 2^n - 1 is Prime, are Granville Numbers. So, there are infinitely many Granville Numbers and the infinite family has 2 prime factors- 2 and a Mersenne Prime. Others include 126, 5540590, 9078520, 22528935, 56918394 and 246650552 having 3, 5, 5, 5, 5 and 5 prime factors.
S-abundant numbers
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:
\sumd\mid{n, d<n, d\inl{S}}d>{n}
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, ...
Examples
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.
Notes and References
- De Koninck JM, Ivić A. On a Sum of Divisors Problem. Publications de l'Institut mathématique. 1996. 64. 78. 9–20. 27 March 2011.
- Book: de Koninck, Jean-Marie . Jean-Marie De Koninck . J. M. . de Koninck . Those Fascinating Numbers . registration . . Providence, RI . 2008 . 40 . 978-0-8218-4807-4 . 317778112 . 2532459 .
