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%I #36 Oct 22 2022 16:20:11
%S 1,7,25,31,37,73,79,85,97,103,193,241,253,271,313,319,337,343,361,517,
%T 553,661,703,721,727,733,745,751,781,799,805,865,925,943,967,1015,
%U 1039,1081,1087,1633,1687,1705,1837,1981,2125,2137,2143,2185,2191,2233,2257,2263,2341,2581,2593,2605,2641,2719,2761,2797,2815,2833,2857,2887,2893,2911,3127
%N Integers n such that the digit set D = (0, 1, -n) used in base 3 expansions of the form Sum_{ -N < j < oo} d_j 3^{-j}, all d_j in D, can represent all real numbers.
%C All nonnegative reals can be represented with ternary digits 0, 1, 2. If you're not allowed to use 2, then you only get something like the Cantor set. But you're back in business if you're allowed to use 0, 1, -1 - this gives the "balanced" ternary representation (so 1 is in the sequence).
%C The sequence is known to be infinite and irregular and is conjectured to have density zero.
%D J. C. Lagarias, Crystals, Tilings and Packings, Hedrick Lectures, Math. Assoc. America MathFest, 2005.
%H Joerg Arndt, <a href="/A110081/b110081.txt">Table of n, a(n) for n = 1..1000</a>
%H David W. Matula, <a href="http://dx.doi.org/10.1145/322344.322355">Basic digit sets for radix representation</a>, J. Assoc. Comput. Mach. 29 (1982), 1131-1143.
%H Don Reble, <a href="/A110081/a110081.txt">Python Program</a>
%e 13/18 = 0.122111111111... in ternary which can't be represented without the 2's. But it is 10.x0111111111... if x = -7: 3 + 0 + (-7)/3 + 1/3^3 + 1/3^4 + 1/3^5 + ... = 3 - 7/3 + (1/27)/(1-(1/3)) = 13/18.
%K nonn,base,nice
%O 1,2
%A _N. J. A. Sloane_, based on correspondence from _R. K. Guy_ and Jeff Lagarias, Aug 31 2005
%E More terms using _Don Reble_'s program from _Joerg Arndt_, Sep 17 2017
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