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Harmonic Harness (Posted on 2013-11-27) Difficulty: 4 of 5
Three positive integers are chosen at random without replacement from 1,2,....,72. What is the probability that the numbers chosen are in harmonic sequence?

Order of choice doesn't matter. For example, 6-3-2 would qualify as numbers in harmonic sequence.

Bonus Question:

Generalize this result (in terms of n) covering the situation where three positive integers are chosen at random without replacement from 1,2,...,n.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts Some thoughts on the bonus Comment 2 of 2 |
Note from Charlie's simulation the values of n at which the (unreduced) numerator of the probability changes.  These are 6, 12, 15, 18, 20...

This happens to be the sequence of integers having divisors d and e with d < e < 2d.  See

A little algebra shows that given integers a < b < c, then they are in harmonic sequence if c = ab / (2a - b).  The denominator implies that a < b < 2a.  

I haven't had any time to take this train of thought further but I obviously wonder if there's a connection, and if it would shed any light on how to compute the probability for a given n. 

e.g. we can always calculate the denominator of the probability, it's just n choose 3.  And the numerator only increases at the values of n on the integer sequence linked above... but as I said, that's as far as I've been able to go and won't be able to come back to it until later.  Perhaps someone else can run with it.

  Posted by tomarken on 2014-02-28 18:49:43
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