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Re: Reciprocals of integers summing to 1
Posted:
Nov 21, 2012 11:49 PM
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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!
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> > 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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> Admittedly no comment was made on permutations of solutions,
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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.
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