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Power Floors


The sequence {|_(3/2)^n_|} is given by 1, 1, 2, 3, 5, 7, 11, 17, 25, 38, ... (OEIS A002379). The first few composite |_(3/2)^n_| occur for n=8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, ... (OEIS A046037), corresponding to the composites 25, 38, 57, 86, 129, 194, 291, 437, 656, ... (OEIS A070758). Similarly, the first few prime |_(3/2)^n_| occur for n=3, 4, 5, 6, 7, 21, 22, 98, ... (OEIS A070759), corresponding to the primes 2, 3, 5, 7, 11, 17, 4987, 7481, 180693856682317883, ... (OEIS A067904).

The sequence {|_(4/3)^n_|} is given by 1, 1, 2, 3, 4, 5, 7, 9, 13, 17, 23, ... (OEIS A064628). The first few composite |_(4/3)^n_| occur for n=5, 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, ... (OEIS A046038), corresponding to composites 4, 9, 42, 56, 74, 99, 133, 177, 236, ... (OEIS A070761). Similarly, the first few prime |_(4/3)^n_| occur for n=4, 6, 7, 9, 10, 11, 12, 38, 42, 59, 96,... (OEIS A070762), corresponding to the primes 2, 3, 5, 7, 13, 17, 23, 31, 55933, 176777, 23517191, ... (OEIS A067905).

There are infinitely many integers of the form |_(3/2)^n_| and |_(4/3)^n_| which are composite, where |_x_| is the floor function (Forman and Shapiro, 1967; Guy 1994, p. 220).


See also

Floor Function, Power, Power Ceilings, Power Floor Prime Sequence, Power Fractional Parts

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References

Forman, W. and Shapiro, H. N. "An Arithmetic Property of Certain Rational Powers." Comm. Pure Appl. Math. 20, 561-573, 1967.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.Sloane, N. J. A. Sequences A002379/M0666, A046037, A046038, A064628, A067904, A067905, A070758, A070759, A070761, and A070762 in "The On-Line Encyclopedia of Integer Sequences."

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Power Floors

Cite this as:

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

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