A321318 - OEIS

%I #20 Jan 29 2020 08:09:17

%S 1,2,2,3,3,3,3,4,4,5,5,5,5,5,5,5,5,7,7,7,7,8,8,7,7,9,9,8,8,7,7,6,6,9,

%T 9,9,9,10,10,9,9,9,9,12,12,13,13,9,9,12,12,12,12,12,12,12,12,14,14,12,

%U 12,11,11,7,7,11,11,12,12,13,13,12,12,15,15,15

%N Number of distinct values obtained by partitioning the binary representation of n into consecutive blocks, and then summing the numbers represented by the blocks.

%H Rémy Sigrist, <a href="/A321318/b321318.txt">Table of n, a(n) for n = 1..16384</a>

%H Elwyn Berlekamp and Joe P. Buhler, <a href="http://www.msri.org/attachments/media/news/emissary/EmissaryFall2011.pdf">Puzzle 6</a>, Puzzles column, Emissary, MSRI Newsletter, Fall 2011, Page 9, Problem 6.

%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.

%H Rémy Sigrist, <a href="/A321318/a321318.gp.txt">PARI program for A321318</a>

%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 are 5 distinct values.

%o (PARI) See Links section

%Y Cf. A321319, A321320, A321321.

%K nonn,look,base

%O 1,2

%A _Jeffrey Shallit_, Nov 04 2018