Numbers: Their Tales, Types, and Treasures.

Chapter 9: Number Relationships

9.3.OTHER TYPES OF AMICABILITY

There are other types of numbers that also have an amicable relationship, such as imperfectly-amicable numbers—two numbers the sums of whose proper divisors are equal. For example, the numbers 20 and 38 are considered imperfectly-amicable numbers, since the proper divisors of 20 are 1, 2, 4, 5, 10, whose sum is 22, and the proper divisors of 38 are 1, 2, 19, whose sum is also 22. Another pair of imperfectly-amicable numbers are 69 and 133, since each has a sum of proper divisors equal to 27. You might want to verify that the numbers 45 and 87 are also imperfectly-amicable numbers.

We can always look for other nice relationships between numbers. Some of them are truly mind-boggling! Take for example, the pair of numbers 6,205 and 3,869, which we will call structurally-amicable numbers, where the following relationship exists:

6,205 = 38² + 69²,
62² + 05² = 3,869.

Notice the symmetry of the breakdown of these two given four-digit numbers. The pair of numbers 5,965 and 7,706 has the same relationship:

5,965 = 77² + 06²,
59² + 65² = 7,706.

There are also curious relationships between numbers that tie them together in an amicable way, such as the pair of numbers 244 and 136, which can be linked as follows:

244 = 1³ + 3³ + 6³,
2³ + 4³ + 4³ = 136.

Ambitious readers may seek other forms of number-pair friendliness!