Suggestion for a sequence: weights on a circle

[Brendan McKay]

Edwin has produced strong evidence that there are no solutions when
n is a prime power.  I can't prove that, but it is easy to show that
there are solutions when n is not a prime power.  Every such number
can be written as m*n where m,n are coprime.  Now continue like in 
this example (n=5,m=3):

    Write a permutation of {1,2,...,n} m times:
       5  1  3  4  2  5  1  3  4  2  5  1  3  4  2
    Add 0,n,2n,...,(m-1)n,0,... cyclically:
       5  6 13  4  7 15  1  8 14  2 10 11  3  9 12
    That's a solution.

    Hugo's example for n=15 suggests another way to do the last step:
    Replace the value v in position i by m*v+(i mod m).

Brendan.