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, 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

Offset: 1

Author

N. J. A. Sloane, based on correspondence from R. K. Guy and Jeff Lagarias, Aug 31 2005

Keywords

• nonn
• base
• nice

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.

Examples

			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.
		

References

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

Links

• Joerg Arndt, Table of n, a(n) for n = 1..1000
• David W. Matula, Basic digit sets for radix representation, J. Assoc. Comput. Mach. 29 (1982), 1131-1143.
• Don Reble, Python Program

Extensions

More terms using Don Reble's program from Joerg Arndt, Sep 17 2017