%I #7 Nov 24 2019 10:00:18

%S 0,8,12,14,17,24,27,28,35,36,39,47,49,51,54,57,61,70,73,78,80,99,122,

%T 130,156,175,184,189,190,198,204,207,208,215,216,226,228,235,243,244,

%U 245,261,271,283,295,304,313,321,322,336,352,367,375,378,379,380,386

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

%C 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.

%C 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.

%H Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.

%e The sequence of terms together with their binary expansions begins:

%e 0:

%e 8: 1000

%e 12: 1100

%e 14: 1110

%e 17: 10001

%e 24: 11000

%e 27: 11011

%e 28: 11100

%e 35: 100011

%e 36: 100100

%e 39: 100111

%e 47: 101111

%e 49: 110001

%e 51: 110011

%e 54: 110110

%e 57: 111001

%e 61: 111101

%e 70: 1000110

%e 73: 1001001

%e 78: 1001110

%e 80: 1010000

%e 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) -> ().

%e 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) -> ().

%t runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;

%t degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;

%t Select[Range[0,100],#==0||runsres[IntegerDigits[#,2]]==degdep[IntegerDigits[#,2]]&]

%Y Positions of 0's in A329867.

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

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

%Y Runs-resistance of binary expansion is A318928.

%Y Cuts-resistance of binary expansion is A319416.

%Y Compositions counted by runs-resistance are A329744.

%Y Compositions counted by cuts-resistance are A329861.

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

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 23 2019