Abundant Numbers

Today's numbered day is 24762 and one of its claims to fame, according the OEIS, is its membership in A228964: smallest sets of 7 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed. The abundant numbers forming this arithmetic sequence are 24762 | 24768 | 24774 | 24780 | 24786 | 24792 | 24798. The next abundant number is 24800 which breaks the pattern.

It might be appropriate in this post to remind myself what constitutes an abundant number and to list some interesting facts about them. To begin, a definition from Wikipedia:

In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example. 

Definition: a number \(n\) for which the sum of divisors \(\sigma(n)>2n \), or, equivalently, the sum of proper divisors (or aliquot sum) \( \text{s}(n)>n\). 

Abundance is the value \( \sigma(n)-2n\) (or \( \text{s}(n)-n\)).

Some of the interesting facts about abundant numbers are:

• the smallest odd abundant number is 945
• The smallest abundant number not divisible by 2 or by 3 is 5391411025
• infinitely many even and odd abundant numbers exist
• every integer greater than 20161 can be written as the sum of two abundant numbers 
• every multiple (beyond 1) of a perfect number is abundant
• every multiple of an abundant number is abundant
• an abundant number with abundance 1 is called a quasiperfect number, although none have yet been found

ADDENDUM: Friday, June 12th 2020

I came across some further interesting facts about abundant numbers on this site. Here is what was mentioned:

• There are at least 10000 pairs of known consecutive abundant integers.
  See A096399 and this file by T. D. Noe.
  
• The triple 171078830, 171078831, 171078832 was apparently found by Laurent Hodges and Michael Reid in 1995.
  
• There are at least 1000 triples of consecutive abundant numbers.
  See A096536 and this file by Donovan Johnson.
  
• The starting term of the smallest consecutive 4-tuple of abundant numbers is at most:
  141363708067871564084949719820472453374 (39 digits)
  by Bruno Mishutka, November 1st 2007. See A094628.