Numbers n for which a sum of n-th roots of unity is a new nonzero minimum (in absolute value).
2, 5, 7, 8, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 35, 41, 43, 47, 53, 57, 59, 61, 67, 71, 73, 77, 79
1
That is, we select a sum that is a real number. If complex absolute value is used, the same numbers appear. The plot of 11-th roots below shows the great symmetry of the sums.
T. D. Noe, Plot of 28 terms
T. D. Noe, Plot sums of 11-th roots of unity with absolute value less than 1
Eric. W. Weisstein, MathWorld: Root of Unity
For example, for 8-th roots of unity numbered r(0) to r(7), abs(r(0) + r(3) + r(5)) = abs(1 + (-1+i)/sqrt(2) + (-1-i)/sqrt(2)) = 1 - sqrt(2) = 0.41421356237…, which is A188582.
(Mma) mnVal = 2; t = {}; Do[rts = N[Roots[x^n == 1, x], 25]; r = Table[rts[[i, 2]], {i, n}]; r2 = Subsets[r, n - 1]; r3 = Abs[Total /@ r2]; s = Sort[r3]; While[s[[1]] == 0, s = Rest[s]]; If[s[[1]] < mnVal, mnVal = s[[1]]; AppendTo[t, n]], {n, 2, 15}]; t
nonn,hard,more
T. D. Noe, Dec 01 2014