Friendship among numbers | Tangente

200 has a friend! --------------

Numbers, too, can have friends: two integers are friends if they have the same abundancy index. An integer's abundancy index is the sum of its divisors divided by the integer itself.

The divisors of 200 are 1, 2, 4, 5, 10, 20, 40, 50, 100 and 200, so its abundancy index is (1 + 2 + 4 + 5 + 8 + 10 + 20 + 25 + 40 + 50 + 100 + 200)/200, or 93/40.

Now, 80 has the same abundancy index. Its divisors are 1, 2, 4, 5, 8, 10, 16, 20, 40 and 80, so its abundancy index is (1 + 2 + 4 + 5 + 8 + 10 + 16 + 20 + 40 + 80)/80, again giving 93/40.

The integers 200 and 80 are therefore friends! They have no other friends below 10⁸.