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Topic: Reciprocals of integers summing to 1
Replies: 21   Last Post: Nov 23, 2012 8:44 PM

 Messages: [ Previous | Next ]
 David Petry Posts: 1,104 Registered: 12/8/04
Re: Reciprocals of integers summing to 1
Posted: Nov 21, 2012 11:49 PM

On Sunday, November 18, 2012 4:45:05 AM UTC-8, Bill Taylor wrote:
> Quite clearly a lot of respondents didn't seem to read the question!
>
>
>

> > For each n, what are the solutions in positive integers
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> > to (1/X1)+(1/X2) + . . . + (1/Xn)=1 ?
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>
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> but in such contexts they are almost always regarded as the same.
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> Clearly repeats among the x_i are allowed.
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> (Though one could also answer with them disallowed.)
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> But most important, A FUNCTION OF n IS REQUIRED.
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> i.e. what is f(n) = card({x1, x2, ... , xn} | etc)
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> where the curly brackets denote unordered multisets.
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> So far we have f(1) = 1, f(2) = 1, f(3) = 3 (seemingly)
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> and no great effort on any higher values.
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> If no-one can get anywhere much theoretically,
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> can some computer whizz at least produce a list of
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> the first several values of f please?

Clearly it grows quite fast.

For n = 4, we have the following sets.

2 3 7 42
2 3 8 24
2 3 9 18
2 3 10 15
2 3 12 12
2 4 5 20
2 4 6 12
2 4 8 8
2 5 5 10
2 6 6 6
3 3 4 12
3 3 6 6
3 4 4 6
4 4 4 4

So f(4) = 14 (I hope)

That's by hand, so I'm not going any further.

If someone does tabulate a few more values, it would probably be a new entry for that list of integer sequences someone has on the internet.

Date Subject Author
11/16/12 Charlie-Boo
11/16/12 William Elliot
11/16/12 Charlie-Boo
11/16/12 William Elliot
11/17/12 Charlie-Boo
11/18/12 Bill Taylor
11/21/12 David Petry
11/22/12 Bill Taylor
11/22/12 Luis A. Rodriguez
11/22/12 David Petry
11/22/12 David Petry
11/23/12 Bill Taylor
11/23/12
11/23/12 Bill Taylor
11/16/12 Don Redmond
11/16/12 gus gassmann
11/16/12 billh04
11/17/12 Luis A. Rodriguez
11/17/12 Charlie-Boo
11/19/12 Luis A. Rodriguez
11/20/12 doumin
11/22/12 Bill Taylor