A321321 Numbers n for which the "partition-and-add" operation applied to the binary representation of n results in only one power of 2.

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 17, 19, 21, 24, 25, 28, 31, 33, 35, 37, 41, 42, 48, 49, 56, 65, 67, 69, 73, 81, 87, 96, 97, 112, 129, 131, 133, 137, 145, 161, 167, 192, 193, 224, 257, 259, 261, 265, 273, 289, 321, 384, 385, 448, 513, 515, 517, 521, 529, 545

Offset: 1

Author

Jeffrey Shallit, Nov 04 2018

Keywords

• nonn

Comments

Conjecture: With the exception of a(1) = 1 and a(17) = 31, all terms have a binary weight of 2 or 3. - Peter Kagey, Jun 14 2019

Examples

			For n = 13, we can partition its binary representation as follows (showing partition and sum of terms): (1101):13, (1)(101):6, (11)(01):4, (110)(1):7, (1)(1)(01):3, (1)(10)(1):4, (11)(0)(1):4, (1)(1)(0)(1):3.  Thus there is only one possible power of 2, namely 4.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..200
• E. Berlekamp, J. Buhler, Puzzle 6, Puzzles column, Emissary Fall (2011) 9.
• Steve Butler, Ron Graham, and Richard Stong, Collapsing numbers in bases 2, 3, and beyond, in The Proceedings of the Gathering for Gardner 10 (2012).
• Steve Butler, Ron Graham, and Richard Strong, Inserting plus signs and adding, Amer. Math. Monthly 123 (3) (2016), 274-279.

Crossrefs

Cf. A321318, A321319, A321320.