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A329868 Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.

%I A329868 #8 Nov 24 2019 10:00:44
%S A329868 0,1,2,7,11,15,18,31,63,75,127,255,511,1023,1234,2047,4095,8191,9638,
%T A329868 16383,32767,65535,131071,262143,524287,1048575,2097151,4194303,
%U A329868 8388607
%N A329868 Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.
%C A329868 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 A329868 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 A329868 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e A329868 The sequence of terms together with their binary expansions begins:
%e A329868       0:
%e A329868       1:                1
%e A329868       2:               10
%e A329868       7:              111
%e A329868      11:             1011
%e A329868      15:             1111
%e A329868      18:            10010
%e A329868      31:            11111
%e A329868      63:           111111
%e A329868      75:          1001011
%e A329868     127:          1111111
%e A329868     255:         11111111
%e A329868     511:        111111111
%e A329868    1023:       1111111111
%e A329868    1234:      10011010010
%e A329868    2047:      11111111111
%e A329868    4095:     111111111111
%e A329868    8191:    1111111111111
%e A329868    9638:   10010110100110
%e A329868   16383:   11111111111111
%e A329868   32767:  111111111111111
%e A329868   65535: 1111111111111111
%t A329868 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t A329868 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A329868 das=Table[If[n==0,0,runsres[IntegerDigits[n,2]]-degdep[IntegerDigits[n,2]]],{n,0,1000000}];
%t A329868 Table[Position[das,i][[1,1]]-1,{i,First/@Gather[das]}]
%Y A329868 Sorted positions of first appearances in A329867.
%Y A329868 Compositions with runs-resistance equal to cuts-resistance are A329864.
%Y A329868 Runs-resistance of binary expansion is A318928.
%Y A329868 Cuts-resistance of binary expansion is A319416.
%Y A329868 Compositions counted by runs-resistance are A329744.
%Y A329868 Compositions counted by cuts-resistance are A329861.
%Y A329868 Binary words counted by runs-resistance are A319411 and A329767.
%Y A329868 Binary words counted by cuts-resistance are A319421 and A329860.
%Y A329868 Cf. A000975, A003242, A107907, A164707, A329738, A329865, A329866.
%K A329868 nonn,more
%O A329868 1,3
%A A329868 _Gus Wiseman_, Nov 23 2019