A sequence similar to Mills can be built based on the powers of 2.
f(n) = floor(C^2^n).
OEIS A059784: 2, 5, 29, 853, 727613, 529420677791, 280286254072681840639693, ...
and its sequence of offsets from the square of the last term:
OEIS A286682: 1, 4, 12, 4, 22, 12, 114, 4, 138, 142, 2956, 6388, 5248, 17532, 96930, 83782, 1464, 897448, 300832, 26908, ...

To be completed...