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Number of powers of 3 whose binary representation contains exactly n 1's.
2

%I #4 Aug 02 2023 13:50:36

%S 1,2,1,1,1,3,0,1,1,1,2,0,1,3,1,1,2,1,1,1,0,1

%N Number of powers of 3 whose binary representation contains exactly n 1's.

%C Number of numbers k >= 0 such that A011754(k) = n.

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

%C After a(22), the sequence undoubtedly continues 0, 1, 3, 2, 1, 1, 1, 1, 0, 2, 1, 4, 1, 1, 0, 2, 4, 1, 2, 3, 0, 0, 2, 1, 1, 1, 1, 0, ..., but there seem to be proofs only for the first 22 terms (Dimitrov and Howe).

%H Vassil S. Dimitrov and Everett W. Howe, <a href="https://arxiv.org/abs/2105.06440">Powers of 3 with few nonzero bits and a conjecture of Erdős</a>, arXiv:2105.06440 [math.NT], 2021.

%H H. G. Senge and E. G. Straus, <a href="https://doi.org/10.1007/BF02018464">PV-numbers and sets of multiplicity</a>, Periodica Mathematica Hungarica 3 (1973), 93-100.

%e There are a(6) = 3 powers of 3 that have exactly 6 binary 1's: 3^5 (11110011 in binary), 3^6 (1011011001), and 3^8 (1100110100001).

%e There is no power of 3 with exactly 7 binary 1's, so a(7) = 0.

%Y Cf. A011754.

%K nonn,base,more

%O 1,2

%A _Pontus von Brömssen_, Jul 31 2023