a/b + b/c + c/a = n

[all at abouthugo.de]

SeqFans,

the following problem came up in the sci.math NG,
with a partially wrong contribution from me. Is there a nice hidden
sequence?

<<
Problem: a/b + b/c + c/a = n
Author: Marcus <gauss202 at yahoo.com> 
Date Posted: Jul 11 2003 11:36:14:000AM

On 11 Jul 2003, Hugo Pfoertner wrote:
>On 10 Jul 2003, Marcus wrote:
>>I'm stuck on this problem that was posed on another message board. The problem is to find which positive integers n can be the result of a/b + b/c + c/a, where a, b, and c are positive integers.
>>
>>Marcus
>
>With a few lines of code I get for the range a,b,c [1..3000]:
>(only first 3 occurrences printed:)
>
> n    a    b    c      a/b         b/c         c/a
> 3    1    1    1   1.0000000   1.0000000   1.0000000
> 5    1    2    4   0.5000000   0.5000000   4.0000000
> 3    2    2    2   1.0000000   1.0000000   1.0000000
> 5    2    4    1   0.5000000   4.0000000   0.5000000
> 5    2    4    8   0.5000000   0.5000000   4.0000000
> 6    2   12    9   0.1666667   1.3333333   4.5000000
>41    2   36   81   0.0555556   0.4444444  40.5000000
> 3    3    3    3   1.0000000   1.0000000   1.0000000
> 6    3   18    4   0.1666667   4.5000000   1.3333333
>66    3  126  196   0.0238095   0.6428571  65.3333333
> 6    4    3   18   1.3333333   0.1666667   4.5000000
>41    4    9  162   0.4444444   0.0555556  40.5000000
>41    4   72  162   0.0555556   0.4444444  40.5000000
>19    5  225   81   0.0222222   2.7777778  16.2000000
>66    6  252  392   0.0238095   0.6428571  65.3333333
>66    9   14  588   0.6428571   0.0238095  65.3333333
>19    9  405   25   0.0222222  16.2000000   2.7777778
>19   10  450  162   0.0222222   2.7777778  16.2000000
> 9   12   63   98   0.1904762   0.6428571   8.1666667
> 9   18   28  147   0.6428571   0.1904762   8.1666667
> 9   24  126  196   0.1904762   0.6428571   8.1666667
>14   28  637  338   0.0439560   1.8846154  12.0714286
>53   28 1323 1458   0.0211640   0.9074074  52.0714286
>14   52 1183   98   0.0439560  12.0714286   1.8846154
>14   56 1274  676   0.0439560   1.8846154  12.0714286
>53   56 2646 2916   0.0211640   0.9074074  52.0714286
>10  175  882 1620   0.1984127   0.5444444   9.2571429
>10  245  450 2268   0.5444444   0.1984127   9.2571429
>10  450 2268  245   0.1984127   9.2571429   0.5444444
> 7  492  695 2981   0.7079137   0.2331432   6.0589431
> 4  535 1381 1408   0.3874004   0.9808239   2.6317757
> 4  611 1951 1403   0.3131727   1.3905916   2.2962357
> 4  631 1558 1685   0.4050064   0.9246291   2.6703645
> 7  695 2981  492   0.2331432   6.0589431   0.7079137
>53 1323 1458   28   0.9074074  52.0714286   0.0211640
> 7 2981  492  695   6.0589431   0.7079137   0.2331432
>
>Number of ocurrences of sum n in 1<=a,b,c<=3000
>          n           #
>          3        3000
>          4          12
>          5        2250
>          6        1248
>          7           3
>          9         150
>         10           6
>         14          18
>         19          60
>         41         198
>         53           6
>         66          60
>
>Hugo Pfoertner

Some of these are not correct. Like the (n,a,b,c) = (4,535,1381,1408) 
one. That must be the result of some kind of rounding error in the 
program. Also some of these solutions are essentially equivalent to 
one another, like any a = b = c will yield n = 3. But yes, there do 
seem to be a lot of possible values for n. But I don't think all 
positive integers are possible.

Marcus
>>

I had set the threshold for accepting a sum
being equal to the nearest integer to 10^(-9)
and the 3 wrong solutions (=4) all had absolute
differences < 10^-9. Can someone with access to
higher precision software improve my results?

Hugo