## Chapter 9. Amicable numbers

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

(February 02, 2008) [Japanese]
### 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 10^{10} by exhaustive search.

### Contents

- Sum of divisors
- Amicable numbers
- Quasi-amicable numbers
- 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 10*^{10}, 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

E-mail : kc2h-msm@asahi-net.or.jpHisanori Mishima