>>13803828An abundant number is a number where the sum of all divisors (including the trivial ones) exceeds two times the number itself. So 12 is an abundant number since 1+2+3+4+6+12=28 which is more than 12*2. 20 is also abundant since 1+2+4+5+10+20=42 and 42 is larger than 2*20. The sum of all divisors is often called the ?-function so if ?(n) > 2n, n is said to be abundant. Or put in another way, if ?(n)/n > 2, it's abundant.
However the limit 2 is somewhat arbitrarily and if anything it would actually make more sense to set the limit at ?2/6 actually. Anyway, the abundancy index is simply ?(n)/n. If we then take e^(?(n)/n) and then floor it, we get a more natural "grouping" of integers abundancy index.
Numbers that consist of just the right combination of primes, will get a very high abundancy index. You want to have many small primes in the prime factorization. But there is also diminishing return of having many of a specific prime. So for instance 64 does not have a very high abundancy index as it's prime factorization is [2 2 2 2 2 2]. There is simply quickly diminishing returns on adding twos. We are looking for a specific pattern to choice the primes in a certain order to maximize the abundancy index as quickly as possible. If we start off by 1 (the void of primes) the optimal order for choosing the first 36 primes are
2,3,2,5,2,3,7,2,11,13,2,3,5,17,19,23,2,29,31,7,3,37,41,43,2,47,53,59,5,61,67,71,73,11,79,2
