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Power Ceilings

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Consider the sequence {x_n}_(n=0)^infty defined by x_0=1 and

x_(n+1)=[3/2x_n],

where [z] is the ceiling function. For n=0, 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 x_n satisfies

x_n=|_K(3/2)^n_|

for all n, where K=1.6222705028... (OEIS A083286) is analogous to Mills' constant in the sense that the formula is useless unless K is known exactly ahead of time (Odlyzko and Wilf 1991, Finch 2003).

See also

Ceiling Function, Power Floors, Power Fractional Parts

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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.

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Power Ceilings

Cite this as:

Weisstein, Eric W. "Power Ceilings." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PowerCeilings.html

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