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Friendly numbers (Posted on 2023-06-02) Difficulty: 4 of 5
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.

No Solution Yet Submitted by Jer    
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Comments: ( You must be logged in to post comments.)
  Subject Author Date
Part 1 Puzzle Thoughts K Sengupta2024-02-23 22:40:10
Hint for task 2Jer2023-06-03 09:34:37
my Task 2 failed attemptLarry2023-06-02 12:45:30
#1 and a start at #4 and failed attempt at 2Charlie2023-06-02 12:13:13
Task 1 onlyLarry2023-06-02 11:50:34
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