A346497 List of powers of 2 written in base 3 which contain no zero digits.

1, 2, 11, 22, 121, 1122221122

Offset: 1

Author

Rafael Castro Couto, Jul 20 2021

Keywords

• nonn
• base
• hard
• more

Comments

The listed terms are the base-3 expansions of 1, 2, 4, 8, 16, and 32768.

The program shows that there are no other terms less than 2^1000.

a(7) > 2^(10^7). - Martin Ehrenstein, Jul 27 2021

If it exists, a(7) > 2^(10^21). - Robert Saye, Mar 23 2022

References

• David Wells, "The Penguin Dictionary of Curious and Interesting Numbers" (1997), p. 123.

Links

• R. C. Couto, Number of nonzero digits in 2^n base 3
• Robert I. Saye, On two conjectures concerning the ternary digits of powers of two, arXiv:2202.13256 [math.NT], 2022; J. Integer Seq. 25 (2022) Article 22.3.4.

Crossrefs

Cf. A102483, A004642 (all powers of 2 in base 3), A104320 (number of zeros in ternary representation of 2^n), A130693 (same problem in base 10).

Programs

Mathematica

pwr = 1; Do[pwr = Mod[2*pwr, 3^100]; d = Union[IntegerDigits[pwr, 3]]; If[Intersection[d, {0}] == {}, Print[IntegerString[pwr, 3]]], {n, 10000000}] (* Ricardo Bittencourt, Jul 07 2021 *)
Select[Table[FromDigits[IntegerDigits[2^n,3]],{n,0,100}],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Feb 18 2025 *)

Formula

a(n) = A007089(2^A102483(n)). - Michel Marcus, Jul 23 2021