A130693 -id:A130693 - cp's OEIS frontend

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

1, 2, 11, 22, 121, 1122221122

Offset: 1

Rafael Castro Couto, Jul 20 2021

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

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

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.

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).

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 *)

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

A139126 Least k such that the last n decimal digits of 2^k are all powers of 2.

Original entry on oeis.org

0, 7, 7, 18, 19, 90, 91, 271, 1751, 18807, 56589, 56589, 56589, 56589, 899791, 899791, 2814790, 7635171, 7635171, 39727671, 99530619, 233093807, 233093807, 233093807, 233093807

Offset: 1

T. D. Noe, Apr 08 2008

Does k exist for all n? This sequence is inspired by A130693, which lists all known powers of 2 whose digits are all powers of 2 (that is, 1, 2, 4, or 8).

			2^19=524288 is the least power of 2 ending with 5 digits that are powers of 2.

k=1; Join[{0}, Table[k--; pwr=PowerMod[2,k,10^n]; While[k++; pwr=Mod[2*pwr,10^n]; d=Union[IntegerDigits[pwr,10,n]]; Intersection[d,{3,5,6,7,9,0}]!={}]; k, {n,2,10}]]

cp's OEIS Frontend

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

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