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A329868
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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.
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5
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0, 1, 2, 7, 11, 15, 18, 31, 63, 75, 127, 255, 511, 1023, 1234, 2047, 4095, 8191, 9638, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The sequence of terms together with their binary expansions begins:
0:
1: 1
2: 10
7: 111
11: 1011
15: 1111
18: 10010
31: 11111
63: 111111
75: 1001011
127: 1111111
255: 11111111
511: 111111111
1023: 1111111111
1234: 10011010010
2047: 11111111111
4095: 111111111111
8191: 1111111111111
9638: 10010110100110
16383: 11111111111111
32767: 111111111111111
65535: 1111111111111111
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MATHEMATICA
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runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
das=Table[If[n==0, 0, runsres[IntegerDigits[n, 2]]-degdep[IntegerDigits[n, 2]]], {n, 0, 1000000}];
Table[Position[das, i][[1, 1]]-1, {i, First/@Gather[das]}]
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CROSSREFS
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Sorted positions of first appearances in A329867.
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.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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