Friendly Numbers: What could possibly make two numbers friendly?

Well, mathematicians have decided that two numbers are considered friendly (or as often used in the more sophisticated literature, “amicable”) if the sum of the proper divisors of one equals the second and the sum of the proper divisors of the second number equals the first number.

Have a look at the smallest pair of friendly numbers: 220 and 284.

The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Their sum is 1+2+4+5+10+11+20+22+44+55+110= 284.

The proper divisors of 284 are 1, 2, 4, 71, and 142, and their sum is 1 + 2 + 4 + 71 + 142 = 220.

This shows the two numbers are friendly numbers.

The second pair of friendly numbers to be discovered (by Pierre de Fermat, 1601–1665) was 17,296 and 18,416:

17,296 = 2⁴.23.47 and 18,416 = 2⁴.1,151

The sum of the proper factors of 17,296 is

1+2+4+8+16+23+46+47+92+94+184+188+368+376+752+1081+2162+4324+8648=18416

The sum of the proper factors of 18,416 is

1 + 2 + 4 + 8 + 16 + 1,151 + 2,302 + 4,604 + 9,208 = 17,296

Here are a few more friendly pairs of numbers:

1,184 and 1,210

2,620 and 2,924

5,020 and 5,564

6,232 and 6,368

10,744 and 10,856

9,363,584 and 9,437,056

111,448,537,712 and 118,853,793,424

Want to verify the above pairs’ “friendliness”?

Well, the following is one method for finding friendly numbers.

Let

a = 3.2ⁿ − 1

b = 3.2ⁿ⁻¹ − 1

c = 3².2²ⁿ⁻¹ − 1

where n is an integer greater than or equal to 2 and a, b, & c are all prime numbers. Then 2ⁿab and 2ⁿc are friendly numbers.

(Notice that for n ≤ 200, the values of n = 2, 4 and 7 give us a, b, and c to be prime.)