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Numbers n for which the "partition-and-add" operation applied to the binary representation of n results in only one power of 2.
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%I #12 Jun 14 2019 04:02:31

%S 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,

%T 65,67,69,73,81,87,96,97,112,129,131,133,137,145,161,167,192,193,224,

%U 257,259,261,265,273,289,321,384,385,448,513,515,517,521,529,545

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

%C 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

%H Peter Kagey, <a href="/A321321/b321321.txt">Table of n, a(n) for n = 1..200</a>

%H E. Berlekamp, J. Buhler, <a href="http://www.msri.org/attachments/media/news/emissary/EmissaryFall2011.pdf">Puzzle 6</a>, Puzzles column, Emissary Fall (2011) 9.

%H Steve Butler, Ron Graham, and Richard Stong, <a href="http://www.math.ucsd.edu/~ronspubs/mis_17_bases.pdf">Collapsing numbers in bases 2, 3, and beyond</a>, in The Proceedings of the Gathering for Gardner 10 (2012).

%H Steve Butler, Ron Graham, and Richard Strong, <a href="http://orion.math.iastate.edu/butler/papers/16_03_insert_and_add.pdf">Inserting plus signs and adding</a>, Amer. Math. Monthly 123 (3) (2016), 274-279.

%e 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.

%Y Cf. A321318, A321319, A321320.

%K nonn

%O 1,2

%A _Jeffrey Shallit_, Nov 04 2018