The abundancy index of a number is the ratio of the sum of its factors to the number itself.

For example a(12)=σ(n)/n=(1+2+3+4+6+12)/12 = 28/12 = 7/3.

Two numbers n and m are said to be friends if a(n)=a(m). Numbers that have at least one friend are called friendly numbers. Numbers with no friends are called solitary.

Task 1: Find at least one friend for the number 12.

Task 2: Prove there is a set of at least 50 mutual friends.

Task 3: Prove if the fraction σ(n)/n cannot be reduced then n is solitary.

Task 4: a(18)=39/18 which does reduce to 13/6. Prove 18 is solitary nevertheless.

Quite a few numbers have unknown status. The smallest of which is 10. I won't ask you to prove the status of any of these.

You won't be able to brute force task 2.  You are familiar with these numbers, though.  The set as currently known has 51 members of which the largest 3 have over 40 million digits.

Posted by Jer on 2023-06-03 09:34:37