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

1,2

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

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.

Example

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.

Crossrefs

Cf. A321318, A321319, A321320.

Sequence in context: A361386 A361786 A079905 * A154611 A189669 A164028

Adjacent sequences: A321318 A321319 A321320 * A321322 A321323 A321324

Keyword

nonn

Author

Jeffrey Shallit, Nov 04 2018

Status

approved