A329865 Numbers whose binary expansion has the same runs-resistance as cuts-resistance.

0, 8, 12, 14, 17, 24, 27, 28, 35, 36, 39, 47, 49, 51, 54, 57, 61, 70, 73, 78, 80, 99, 122, 130, 156, 175, 184, 189, 190, 198, 204, 207, 208, 215, 216, 226, 228, 235, 243, 244, 245, 261, 271, 283, 295, 304, 313, 321, 322, 336, 352, 367, 375, 378, 379, 380, 386

Offset: 1

Author

Gus Wiseman, Nov 23 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.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

Examples

			The sequence of terms together with their binary expansions begins:
    0:
    8:      1000
   12:      1100
   14:      1110
   17:     10001
   24:     11000
   27:     11011
   28:     11100
   35:    100011
   36:    100100
   39:    100111
   47:    101111
   49:    110001
   51:    110011
   54:    110110
   57:    111001
   61:    111101
   70:   1000110
   73:   1001001
   78:   1001110
   80:   1010000
For example, 36 has runs-resistance 3 because we have (100100) -> (1212) -> (1111) -> (4), while the cuts-resistance is also 3 because we have (100100) -> (00) -> (0) -> ().
Similarly, 57 has runs-resistance 3 because we have (111001) -> (321) -> (111) -> (3), while the cuts-resistance is also 3 because we have (111001) -> (110) -> (1) -> ().
		

Links

• Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Crossrefs

Positions of 0's in A329867.

The version for runs-resistance equal to cuts-resistance minus 1 is A329866.

Compositions with runs-resistance equal to cuts-resistance are A329864.

Runs-resistance of binary expansion is A318928.

Cuts-resistance of binary expansion is A319416.

Compositions counted by runs-resistance are A329744.

Compositions counted by cuts-resistance are A329861.

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

Binary words counted by cuts-resistance are A319421 and A329860.

Cf. A000975, A003242, A107907, A164707, A319420, A329738, A329868.

Programs

Mathematica

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Select[Range[0,100],#==0||runsres[IntegerDigits[#,2]]==degdep[IntegerDigits[#,2]]&]