A364650 Number of powers of 3 whose binary representation contains exactly n 1's.

1, 2, 1, 1, 1, 3, 0, 1, 1, 1, 2, 0, 1, 3, 1, 1, 2, 1, 1, 1, 0, 1

Offset: 1

Author

Pontus von Brömssen, Jul 31 2023

Keywords

• nonn
• base
• more

Comments

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

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

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).

Examples

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

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.