-1,-1,0,0,+1,0,+1,0,0,0,0,-1,......

[Antreas P. Hatzipolakis]

Let c_r be defined for r = 1,2,...., by

c_r = 0, if r is not of the form m(3m+-1)/2,
c_r = (-1)^m, if r is of the form m(3m+-1)/2,

Thus,

r   |  1   2  3  4   5  6   7  8  9 10 11  12
    |
c_r | -1, -1, 0, 0, +1, 0, +1, 0, 0, 0, 0, -1

Show that the sum of the kth powers of the roots of

x^n + c_1x^(n-1) + c_2x^(n-2) + .... + c_(n-1)x + c_n = 0

(for k = 1,2,...,n) is s_k = the sum of the divisprs of k (independent of n).
Thus,

k   | 1 2 3 4 5 6  7  8  9 10 11 12 ...
    |
s_r | 1 3 4 7 6 12 8 15 13 18 12 28 ....

The American Mathematical Monthly 39 (1932), p. 300, #3553 by A. A. Bennett


Antreas