A365215 Largest k such that the binary representation of 3^k has exactly n 1's, or -1 if no such k exists.

0, 2, 4, 3, 7, 8, -1, 9, 10, 12, 16, -1, 11, 18, 15, 24, 20, 25, 22, 21, -1, 23

Offset: 1

Author

Pontus von Brömssen, Aug 26 2023

Keywords

• sign
• base
• more

Comments

Largest k such that A011754(k) = n, or -1 if no such k exists.

Senge and Straus prove that a(n) is finite for all n.

The first 22 terms are from Dimitrov and Howe (2021). After a(22), the sequence conjecturally but very likely continues -1, 26, 30, 32, 36, 40, 34, 27, -1, 39, 49, 45, 53, 38, -1, 47, 56, 57, 50, 58, -1, -1, 66, 51, 67, 59, 62, -1, ... .

Links

• Vassil S. Dimitrov and Everett W. Howe, Powers of 3 with few nonzero bits and a conjecture of Erdős, arXiv:2105.06440 [math.NT], 2021.
• H. G. Senge and E. G. Straus, PV-numbers and sets of multiplicity, Periodica Mathematica Hungarica 3 (1973), 93-100.

Crossrefs

Cf. A011754, A364650, A365214.

Programs

Mathematica

LargestK[n_Integer] := Module[{k = 1000(*Assuming 1000 is large enough for the search.  Adjust if necessary.*), binCount}, While[k >= 0, binCount = Total[IntegerDigits[3^k, 2]]; If[binCount == n, Return[k]]; k--;]; -1]; Table[LargestK[n], {n, 22}] (* Robert P. P. McKone, Aug 26 2023 *)