A365214 -id:A365214 - cp's OEIS frontend

A375472 Least k such that the ternary representation of 2^k has exactly 2*n 1's, or -1 if no such k exists.

1, 2, 8, 14, 24, 26, 42, 45, 50, 53, 70, 74, 96, 76, 124, 98, 116, 121, 143, 141, 179, 150, 187, 181, 192, 215, 209, 233, 220, 257, 245, 264, 243, 278, 260, 310, 297, 303, 315, 339, 329, 387, 341, 357, 354, 366, 403, 420, 350, 400, 411, 415, 474, 455, 466, 442

Offset: 0

Pontus von Brömssen, Aug 17 2024

			For n = 3, the smallest power of 2 with exactly 2*3 = 6 1's in its ternary representation is 2^14 = 211110211_3, so a(3) = 14.

Cf. A020857, A060035, A104321, A365214.

a(n) = my(k=1); while (#select(x->(x==1), digits(2^k, 3)) != 2*n, k++); k; \\ Michel Marcus, Aug 17 2024

Conjecture: a(n) ~ 6*log_2(3)*n = 6*A020857*n.

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 *)

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A375472 Least k such that the ternary representation of 2^k has exactly 2*n 1's, or -1 if no such k exists.

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A365215 Largest 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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