A078608 a(n) = ceiling(2/(2^(1/n)-1)).

2, 5, 8, 11, 14, 17, 20, 23, 25, 28, 31, 34, 37, 40, 43, 46, 49, 51, 54, 57, 60, 63, 66, 69, 72, 75, 77, 80, 83, 86, 89, 92, 95, 98, 100, 103, 106, 109, 112, 115, 118, 121, 124, 126, 129, 132, 135, 138, 141, 144, 147, 150, 152, 155, 158, 161, 164, 167, 170, 173, 176, 178, 181

Offset

1,1

Comments

For n >= 2, a(n) is the least positive integer x such that 2*x^n > (x+2)^n. For example, a(2)=5 as 4^2=16, 5^2=25, 6^2=36 and 7^2=49.

Coincides with floor( 2*n/(log 2) ) for all n from 1 to 777451915729367 but differs at 777451915729368. See A129935.

The first few values of n for which this sequence differs from floor( 2*n/(log 2) ) were found by Dean Hickerson in 2002. - N. J. A. Sloane, Apr 30 2014

The sequence floor( log(n)/(2*log(2)) ) is mentioned by Erdős and Selfridge (1973). This sequence begins 0,0,0,1,1,1,1,... = 0 (3 times), 1 (12 times), 2 (48 times), 3 (192 times), 4 (768 times), ..., and grows too slowly to have its own entry. It is related to the game studied by Hales and Jewett (1963). - N. J. A. Sloane, Dec 02 2016

References

S. Golomb, "Martin Gardner and Tictacktoe," in Demaine, Demaine, and Rodgers, eds., A Lifetime of Puzzles, A K Peters, 2008, pp. 293-301.

S. W. Golomb and A. W. Hales, "Hypercube Tic-Tac-Toe", in "More Games of No Chance", ed. R. J. Nowakowski, MSRI Publications 42, Cambridge University Press, 2002, pp. 167-182. Here it is stated that the first counterexample is at n=6847196937, an error due to faulty multiprecision arithmetic. The correct value was found by Dean Hickerson in 2002, and J. Buhler in 2004, and is reported in S. Golomb (2008).

Dean Hickerson, Email to Jon Perry and N. J. A. Sloane, Dec 16 2002. Gives first three terms of A129935: 777451915729368, 140894092055857794, 1526223088619171207, as well as five later terms. - N. J. A. Sloane, Apr 30 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Max Alekseyev and others, Integer Parts [in Russian]

Max Alekseyev and others, Integer Parts [in Russian] [Cached copy in pdf format]

Art of Problem Solving, Logarithmic Identity

Paul Erdős and John L. Selfridge, On a combinatorial game, Journal of Combinatorial Theory, Series A 14.3 (1973): 298-301.

S. W. Golomb and A. W. Hales, Hypercube Tic-Tac-Toe

A. W. Hales and R. I. Jewett, Regularity and Positional Games, Transactions of the American Mathematical Society, vol. 106, no. 2, Feb. 1963, 222-229.

K. O'Bryant, The sequence of fractional parts of roots, arXiv preprint arXiv:1410.2927 [math.NT], 2014-2015.

Index entries for sequences which agree for a long time but are different

Mathematica

Table[(Ceiling[2/(2^(1/n)-1)]), {n, 1, 100}] (* Vincenzo Librandi, May 01 2014 *)

Prog

(PARI) for (n=2, 50, x=2; while (2*x^n<=((x+2)^n), x++); print1(x", "))
(Haskell)
a078608 = ceiling . (2 /) . (subtract 1) . (2 **) . recip . fromIntegral
-- Reinhard Zumkeller, Mar 27 2015

Crossrefs

Cf. A078607, A078609, A129935.

Sequence in context: A276889 A276877 A329961 * A329988 A189934 A189386

Adjacent sequences: A078605 A078606 A078607 * A078609 A078610 A078611

Keyword

nonn

Author

Jon Perry, Dec 09 2002

Extensions

Edited by Dean Hickerson, Dec 17 2002

Revised by N. J. A. Sloane, Jun 07 2007

Status

approved