A102483 Numbers k such that 2^k contains no zeros in base 3.

0, 1, 2, 3, 4, 15

Offset: 1

Author

N. J. A. Sloane, Feb 25 2005

Keywords

• nonn
• base
• hard

Comments

I conjectured in 1973 that there are no further terms. This question is still open.

A104320(a(n)) = 0. - Reinhard Zumkeller, Mar 01 2005

No other terms less than 200000. - Robert G. Wilson v, Dec 06 2005

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

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

Links

• Robert I. Saye, On two conjectures concerning the ternary digits of powers of two, J. Integer Seq. 25 (2022) Article 22.3.4.
• N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
• Eric Weisstein's World of Mathematics, Ternary

Crossrefs

Cf. A007377, A004642, A346497.

Programs

Mathematica

Select[ Range@1000, FreeQ[ IntegerDigits[2^#, 3], 0] &] (* Robert G. Wilson v, Dec 06 2005 *)

PARI

for (n=0, 100, if (vecmin(digits(2^n, 3)), print1(n, ", "))) \\ Michel Marcus, Mar 25 2015