A228367 n-th element of the ruler function plus the highest power of 2 dividing n.

2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 38, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 71, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7

Offset: 1

Author

Omar E. Pol, Aug 22 2013

Keywords

• nonn
• tabf

Comments

a(n) is also the length of the n-th pair of orthogonal line segments in a diagram of compositions, see example.

a(n) is also the largest part plus the number of parts of the n-th region of the mentioned diagram (if the axes both "x" and "y" are included in the diagram).

a(n) is also the number of toothpicks added at n-th stage to the structure of A228366. Essentially the first differences of A228366.

The equivalent sequence for partitions is A207779.

Examples

			Illustration of initial terms (n = 1..16) using a diagram of compositions in which A001511(n) is the length of the horizontal line segment in row n and A006519(n) is the length of the vertical line segment ending in row n. Hence a(n) is the length of the n-th pair of orthogonal line segments. Also counting both the x-axis and the y-axis we have that A001511(n) is also the largest part of the n-th region of the diagram and A006519(n) is also the number of parts of the n-th region of the diagram, see below.
---------------------------------------------------------
.                Diagram of
n   A001511(n)  compositions   A006519(n)    a(n)
---------------------------------------------------------
1       1        _| | | | |        1          2
2       2        _ _| | | |        2          4
3       1        _|   | | |        1          2
4       3        _ _ _| | |        4          7
5       1        _| |   | |        1          2
6       2        _ _|   | |        2          4
7       1        _|     | |        1          2
8       4        _ _ _ _| |        8         12
9       1        _| | |   |        1          2
10      2        _ _| |   |        2          4
11      1        _|   |   |        1          2
12      3        _ _ _|   |        4          7
13      1        _| |     |        1          2
14      2        _ _|     |        2          4
15      1        _|       |        1          2
16      5        _ _ _ _ _|       16         21
...
If written as an irregular triangle the sequence begins:
  2;
  4;
  2, 7;
  2, 4, 2, 12;
  2, 4, 2, 7, 2, 4, 2, 21;
  2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 38;
  ...
Row lengths is A011782. Right border gives A005126.
Counting both the x-axis and the y-axis we have that A038712(n) is the area (or the number of cells) of the n-th region of the diagram. Note that adding only the x-axis to the diagram we have a tree. - _Omar E. Pol_, Nov 07 2018
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16383
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to toothpick sequences

Crossrefs

Cf. A001511, A001792, A005126, A006519, A011782, A038712, A139250, A139251, A207779, A228366.

Programs

Mathematica

Array[1 + # + 2^# &[IntegerExponent[#, 2]] &, 84] (* Michael De Vlieger, Nov 06 2018 *)

PARI

A228367(n) = (1 + valuation(n,2) + 2^valuation(n,2)); \\ Antti Karttunen, Nov 06 2018

Python

def A228367(n): return (m:=n&-n)+m.bit_length() # Chai Wah Wu, Jul 14 2022

Formula

a(n) = A001511(n) + A006519(n).