A360649 The exponents that occur in the greedy representation of 1/2 as a sum of powers of 2/3.

2, 8, 11, 14, 16, 26, 33, 38, 45, 48, 51, 53, 65, 69, 72, 80, 83, 89, 94, 101, 105, 109, 115, 118, 123, 131, 139, 142, 148, 152, 157, 160, 164, 170, 176, 179, 182, 185, 188, 193, 197, 208, 214, 220, 223, 225, 232, 234, 240, 243, 247, 250, 254, 258, 261, 271

Offset: 1

Author

James Propp, Feb 15 2023

Keywords

• nonn

Comments

These numbers correspond to the 1's in the (3/2)-expansion of 1/2, as defined by Renyi.

Examples

			The first power of 2/3 that is smaller than 1/2 is (2/3)^2, so the first term of the sequence is 2. Subtracting (2/3)^2 from 1/2 leaves 1/18. The first power of 2/3 that is less than 1/18 is (2/3)^8, so the next term of the sequence is 8.
		

Links

• W. Parry, On the beta-Expansions of Real Numbers, Acta Math. Acad. Sci. Hungar. 11, 401-416, 1960.
• A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8 (1957) 477-493.

Crossrefs

Cf. A058840, A077468.

Programs

Maple

x:= 1/2:
for i from 1 to 100 do
  A[i]:= ceil(log[2/3](x));
  x:= x-(2/3)^A[i];
od:
seq(A[i],i=1..100); # Robert Israel, Feb 15 2023

Mathematica

PositionIndex[RealDigits[1/2, 3/2, 100, -1][[1]]][[2]]

Formula

a(n) = A077468(n+1) - 1. - Andrey Zabolotskiy, Nov 03 2024