A364650 -id:A364650 - cp's OEIS frontend

A365214 Least k such that the binary representation of 3^k has exactly n 1's, or -1 if no such k exists.

0, 1, 4, 3, 7, 5, -1, 9, 10, 12, 13, -1, 11, 14, 15, 24, 19, 25, 22, 21, -1, 23

Offset: 1

Pontus von Brömssen, Aug 26 2023

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

The first 22 terms are from Dimitrov and Howe (2021). After a(22), the sequence continues (with the -1's conjectural but very likely correct) -1, 26, 28, 31, 36, 40, 34, 27, -1, 35, 49, 33, 53, 38, -1, 42, 41, 57, 48, 52, -1, -1, 46, 51, 67, 59, 62, -1, ... .

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.

Cf. A011754, A364650, A365215, A375472.

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

Original entry on oeis.org

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

Offset: 1

Pontus von Brömssen, Aug 26 2023

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, ... .

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.

Cf. A011754, A364650, A365214.

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 *)

cp's OEIS Frontend

A365214 Least k such that the binary representation of 3^k has exactly n 1's, or -1 if no such k exists.

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