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The Granville set
- Let and for all let if:
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 that are strictly less than the sum of their divisors in .
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:
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 · 3² · 7 .
Numbers that violate the inequality in the above definition are known as -abundant numbers. That is, the -abundant numbers are the natural numbers that are strictly greater than the sum of their divisors in ; 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, ... (sequence A181487 in OEIS)
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.