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

%I #47 Aug 23 2023 12:18:58

%S 1,2,11,22,121,1122221122

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

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

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

%C a(7) > 2^(10^7). - _Martin Ehrenstein_, Jul 27 2021

%C If it exists, a(7) > 2^(10^21). - _Robert Saye_, Mar 23 2022

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

%H R. C. Couto, <a href="https://codepen.io/rafaelcastrocouto/pen/jOmwxQj">Number of nonzero digits in 2^n base 3</a>

%H Robert I. Saye, <a href="https://arxiv.org/abs/2202.13256">On two conjectures concerning the ternary digits of powers of two</a>, arXiv:2202.13256 [math.NT], 2022; J. Integer Seq. 25 (2022) Article 22.3.4.

%F a(n) = A007089(2^A102483(n)). - _Michel Marcus_, Jul 23 2021

%t 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 *)

%Y 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).

%K nonn,base,hard,more

%O 1,2

%A _Rafael Castro Couto_, Jul 20 2021