Sums of reciprocals of perfect powers

These are cute facts that I didn't know about somehow until just now encountering them on Wikipedia. \sum_{k perfect power, excluding 1} 1/(k-1) = \sum_{n=2}^{\infty}\sum_{m=2}^{\infty} 1/n^m = 1 In the first sum, perfect powers are taken without repeats: e.g., 3^4=9^2 appears only once as k. In the second sum, of course, repeats do occur.