Chapter 9. Amicable numbers
(B4 Amicable numbers, B5 Quasi-amicable or betrothed numbers)

Abstract

Let σ(n) be the sum of divisors of integer n.
For different two numbers m, n, if

σ(m)-m = n
σ(n)-n = m

then these numbers are called as amicable numbers or amicable pairs.
220 and 284 is the smallest example of amicable pair.

This condition is equivalent to

σ(m)=σ(n)=m+n.

If the integer pair of m and n is

σ(m)-m-1 = n
σ(n)-n-1 = m

then these numbers are called as quasi-amicable numbers or quasi-amicable pairs.
48 and 75 is the smallest example of quasi-amicable pair.

This condition is equivalent to

σ(m)=σ(n)=m+n+1.

If the integer pair of m and n is

σ(m)=σ(n)=m+n-1

then these numbers are called as augmented amicable numbers or augmented amicable pairs.
6160 and 11697 is the smallest example of augmented amicable pair.

In this chapter, we try to find all solutions of amicable, quasi-amicable, augmented amicable numbers, up to smaller numbers less than 10¹⁰ by exhaustive search.

Contents

1. Sum of divisors
2. Amicable numbers
3. Quasi-amicable numbers
4. Augmented amicable numbers

Reference

[1] Richard K. Guy, Unsolved Problems in Number Theory (Third Edition), Springer, 2005.
[2] H.J.J.te Riele, Computaion of All the Amicable Pairs Below 10¹⁰, Vol.47, Num.175, July 1986, 361-368
[3] Herman J. J. te Riele, On Generating New Amicable Pairs from Given Amicable Pairs, Mathematics of Computations Vol.42, Num. 165, Jan. 1984, 219-223