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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 (list; graph; refs; listen; history; text; internal format)
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
David W. Matula, Basic digit sets for radix representation, J. Assoc. Comput. Mach. 29 (1982), 1131-1143.
Don Reble, Python Program
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
Sequence in context: A368532 A065660 A100496 * A140716 A141393 A226366
KEYWORD
nonn,base,nice
AUTHOR
N. J. A. Sloane, based on correspondence from R. K. Guy and Jeff Lagarias, Aug 31 2005
EXTENSIONS
More terms using Don Reble's program from Joerg Arndt, Sep 17 2017
STATUS
approved

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Last modified April 14 00:20 EDT 2024. Contains 371652 sequences. (Running on oeis4.)