a(n) = smallest k with sigma(sigma(k)) = n*k (or 0 if no such k exists) is A272930 and has a gaping hole in it: 1, 2, 8, 15, ?, 42, 24, 60, 168, 480, 4404480, 2200380, 57120, 217728, 1058148, 7526400, ?, ... The comments say the search for a(5) has gone out to 5*10^11, and that it is conjectured to be nonzero.