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Baxter-Hickerson numbers.
3

%I #19 Nov 23 2020 17:09:26

%S 2,64037,6634003367,666334000333667,66663334000033336667,

%T 6666633334000003333366667,666666333334000000333333666667,

%U 66666663333334000000033333336666667

%N Baxter-Hickerson numbers.

%C From _Amiram Eldar_, Nov 23 2020: (Start)

%C Named after Lew Baxter and Dean Hickerson.

%C Pegg (1999) conjectured that the sequence of zeroless cubes (A052045) is finite. On April 19, 1999, Hickerson gave the counterexample: if n == 2 (mod 3) and n >= 5, then the cube of (2*10^(5*n) - 10^(4*n) + 17*10^(3*n-1) + 10^(2*n) + 10^n - 2)/3 is zeroless. Three days later, Baxter gave a simpler variation which is valid for all n>=0 and is given in the Formula section. (End)

%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005. See p. 109.

%H Amiram Eldar, <a href="/A052427/b052427.txt">Table of n, a(n) for n = 0..200</a>

%H Lew Baxter, <a href="https://groups.google.com/g/sci.math/c/gX5a49ZCYrc/m/lTSZRKNdjZoJ">Cubes lacking zeros</a>, sci.math newsgroup, April 22, 1999.

%H Ed Pegg, Jr., <a href="https://groups.google.com/g/sci.math/c/wbrYGeKw5y0/m/4g-QljIS6lMJ">Cube conjecture</a>, sci.math newsgroup, April 18, 1999.

%H Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/numbers.html">Fun with Numbers</a>, mathpuzzle websize.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Baxter-HickersonFunction.html">Baxter-Hickerson Function</a>.

%F a(n) = (2*10^(5*n) - 10^(4*n) + 2*10^(3*n) + 10^(2*n) + 10^n + 1)/3 (Baxter, 1999). - _Amiram Eldar_, Nov 23 2020

%p a(0) = 2, and 2^3 = 8 is zeroless.

%p a(1) = 64037, and 64037^3 = 262598918898653 is zeroless.

%t a[n_] := (2*10^(5*n) - 10^(4*n) + 2*10^(3*n) + 10^(2*n) + 10^n + 1)/3; Array[a, 10, 0] (* _Amiram Eldar_, Nov 23 2020 *)

%Y Subsequence of A052044.

%Y Cf. A016789, A051832, A051833, A052045.

%K nonn

%O 0,1

%A _Eric W. Weisstein_

%E Offset changed to 0 by _Amiram Eldar_, Nov 23 2020