

A110081


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.


1



1, 7, 25, 31, 37, 73, 79, 85, 97, 103, 193, 241, 253, 271, 313, 319, 337, 343, 361, 517, 553, 661, 703, 721, 727, 733, 745, 751, 781, 799, 805, 865, 925, 943, 967, 1015, 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
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OFFSET

1,2


COMMENTS

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).
The sequence is known to be infinite and irregular and is conjectured to have density zero.


REFERENCES

J. C. Lagarias, Crystals, Tilings and Packings, Hedrick Lectures, Math. Assoc. America MathFest, 2005.


LINKS



EXAMPLE

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.


CROSSREFS



KEYWORD

nonn,base,nice


AUTHOR



EXTENSIONS



STATUS

approved



