A329870 Runs-resistance of the binary expansion of n without the first digit.

0, 0, 1, 2, 2, 1, 1, 3, 2, 3, 3, 2, 3, 1, 1, 3, 4, 2, 4, 2, 3, 3, 3, 3, 2, 4, 2, 4, 3, 1, 1, 3, 4, 3, 3, 4, 4, 3, 4, 5, 2, 4, 4, 5, 3, 3, 3, 3, 5, 4, 4, 2, 5, 4, 3, 4, 4, 3, 3, 4, 3, 1, 1, 3, 4, 3, 3, 4, 3, 2, 3, 3, 4, 4, 2, 3, 3, 3, 4, 5, 4, 3, 4, 2, 5, 4

Offset: 2

Author

Gus Wiseman, Nov 25 2019

Keywords

• nonn

Comments

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.

Examples

			Minimal representatives with each image are:
    2: (0)
    4: (0,0) -> (2)
    5: (0,1) -> (1,1) -> (2)
    9: (0,0,1) -> (2,1) -> (1,1) -> (2)
   18: (0,0,1,0) -> (2,1,1) -> (1,2) -> (1,1) -> (2)
   41: (0,1,0,0,1) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2)
  150: (0,0,1,0,1,1,0) -> (2,1,1,2,1) -> (1,2,1,1) -> (1,1,2) -> (2,1) -> (1,1) -> (2)
		

Crossrefs

Keeping the first digit gives A318928.

Cuts-resistance is A319420.

Compositions counted by runs-resistance are A329744.

Binary words counted by runs-resistance are A319411 and A329767.

Cf. A107907, A319416, A329860, A329861, A329865, A329867.

Programs

Mathematica

Table[Length[NestWhileList[Length/@Split[#]&,Rest[IntegerDigits[n,2]],Length[#]>1&]]-1,{n,2,100}]