Granville number
In mathematics, specifically number theory, Granville numbers, also known as formula_1-perfect numbers, are an extension of the perfect numbers. The Granville set. In 1996, Andrew Granville proposed the following construction of a set formula_1: A Granville number is an element of formula_1 for which equality holds, that is, formula_4 is a Granville number if it is equal to the sum of its proper divisors that are also in formula_1. Granville numbers are also called formula_1-perfect numbers. General properties. The elements of formula_1 can be -deficient, -perfect, or -abundant. In particular, 2-perfect numbers are a proper  subset of formula_1. S-deficient numbers. Numbers that fulfill the strict form of the inequality in the above definition are known as formula_1-deficient numbers. That is, the formula_1-deficient numbers are the natural numbers for which the sum of their divisors in formula_1 is strictly less than themselves: S-perfect numbers. Numbers that fulfill equality in the above definition are known as formula_1-perfect numbers. That is, the formula_1-perfect numbers are the natural numbers that are equal the sum of their divisors in formula_1. The first few formula_1-perfect numbers are: Every perfect number is also formula_1-perfect . However, there are numbers such as 24 which are formula_1-perfect but not perfect. The only known formula_1-perfect number with three distinct prime factors is 126 = 2 · 3 · 7 . S-abundant numbers. Numbers that violate the inequality in the above definition are known as formula_1-abundant numbers. That is, the formula_1-abundant numbers are the natural numbers for which the sum of their divisors in formula_1 is strictly greater than themselves: They belong to the complement of formula_1. The first few formula_1-abundant numbers are: Examples. Every deficient number and every perfect number is in formula_1 because the  restriction of the divisors sum to members of formula_1 either decreases the divisors sum or leaves it unchanged. The first natural number that is not in formula_1 is the smallest abundant number, which is 12. The next two abundant numbers, 18 and 20, are also not in formula_1. However, the fourth abundant number, 24, is in formula_1 because the sum of its proper divisors in formula_1 is: In other words, 24 is abundant but not formula_1-abundant because 12 is not in formula_1. In fact, 24 is formula_1-perfect - it is the smallest number that is formula_1-perfect but not  perfect. The smallest odd abundant number that is in formula_1 is 2835, and the smallest pair of consecutive numbers that are not in formula_1 are 5984 and 5985.