Power Ceilings

Consider the sequence  defined by  and

where  is the ceiling function. For , 1, ..., the first few terms are 1, 2, 3, 5, 8, 12, 18, 27, 41, 62, ... (OEIS A061419; Wolfram 2002, p. 100, Fig. (b)).

Odlyzko and Wilf (1991) have shown that  satisfies

for all , where  (OEIS A083286) is analogous to Mills' constant in the sense that the formula is useless unless  is known exactly ahead of time (Odlyzko and Wilf 1991, Finch 2003).

REFERENCES:

Finch, S. R. "Powers of 3/2 Modulo One." §2.30.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 194-199, 2003.

Odlyzko, A. M. and Wilf, H. S. "Functional Iteration and the Josephus Problem." Glasgow Math. J. 33, 235-240, 1991.

Sloane, N. J. A. Sequences A061419 and A083286 in "The On-Line Encyclopedia of Integer Sequences."

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 100, 2002.