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.