

A321321


Numbers n for which the "partitionandadd" operation applied to the binary representation of n results in only one power of 2.


5



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
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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), 274279.


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: A174415 A103826 A079905 * A154611 A189669 A164028
Adjacent sequences: A321318 A321319 A321320 * A321322 A321323 A321324


KEYWORD

nonn


AUTHOR

Jeffrey Shallit, Nov 04 2018


STATUS

approved



