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Positive integers whose binary expansion has cuts-resistance 2.
9

%I #5 Nov 24 2019 09:59:53

%S 3,4,6,9,11,12,13,18,19,20,22,25,26,37,38,41,43,44,45,50,51,52,53,74,

%T 75,76,77,82,83,84,86,89,90,101,102,105,106,149,150,153,154,165,166,

%U 169,171,172,173,178,179,180,181,202,203,204,205,210,211,212,213

%N Positive integers whose binary expansion has cuts-resistance 2.

%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 3: 11

%e 4: 100

%e 6: 110

%e 9: 1001

%e 11: 1011

%e 12: 1100

%e 13: 1101

%e 18: 10010

%e 19: 10011

%e 20: 10100

%e 22: 10110

%e 25: 11001

%e 26: 11010

%e 37: 100101

%e 38: 100110

%e 41: 101001

%e 43: 101011

%e 44: 101100

%e 45: 101101

%e 50: 110010

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

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

%Y Positions of 2's in A319416.

%Y Numbers whose binary expansion has cuts-resistance 1 are A000975.

%Y Binary words with cuts-resistance 2 are conjectured to be A027383.

%Y Compositions with cuts-resistance 2 are A329863.

%Y Cuts-resistance of binary expansion without first digit is A319420.

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

%Y Compositions counted by cuts-resistance are A329861.

%Y Cf. A107907, A114901, A164707, A318928, A319411, A329745, A329865, A329866.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 23 2019