A321318 Number of distinct values obtained by partitioning the binary representation of n into consecutive blocks, and then summing the numbers represented by the blocks.

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, 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, 12, 11, 11, 7, 7, 11, 11, 12, 12, 13, 13, 12, 12, 15, 15, 15

Offset

1,2

Links

Rémy Sigrist, Table of n, a(n) for n = 1..16384

Elwyn Berlekamp and Joe P. Buhler, Puzzle 6, Puzzles column, Emissary, MSRI Newsletter, Fall 2011, Page 9, Problem 6.

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.

Rémy Sigrist, PARI program for A321318

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 are 5 distinct values.

Prog

(PARI) See Links section

Crossrefs

Cf. A321319, A321320, A321321.

Sequence in context: A329242 A266193 A114214 * A270362 A196383 A074198

Adjacent sequences: A321315 A321316 A321317 * A321319 A321320 A321321

Keyword

nonn,look,base

Author

Jeffrey Shallit, Nov 04 2018

Status

approved