In 1996, Andrew Granville proposed the following construction of a set
:[1] A Granville number is an element of
is a Granville number if it is equal to the sum of its proper divisors that are also in
Granville numbers are also called
can be k-deficient, k-perfect, or k-abundant.
In particular, 2-perfect numbers are a proper subset of
[1] Numbers that fulfill the strict form of the inequality in the above definition are known as
is strictly less than themselves: Numbers that fulfill equality in the above definition are known as
-perfect number with three distinct prime factors is 126 = 2 · 32 · 7.
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.
Numbers that violate the inequality in the above definition are known as
is strictly greater than themselves: They belong to the complement of
because the restriction of the divisors sum to members of
either decreases the divisors sum or leaves it unchanged.
because the sum of its proper divisors in
The smallest odd abundant number that is in
is 2835, and the smallest pair of consecutive numbers that are not in