A164707 -id:A164707 - cp's OEIS frontend

A329867 Runs-resistance minus cuts-resistance of the binary expansion of n.

0, -1, 1, -1, 1, 1, 1, -2, 0, 1, 1, 2, 0, 2, 0, -3, -1, 0, 3, 2, 2, 1, 3, 1, 0, 2, 2, 0, 0, 1, -1, -4, -2, -1, 2, 0, 0, 3, 2, 0, 1, 3, 1, 2, 1, 2, 2, 0, -1, 0, 1, 0, 2, 2, 0, -1, -1, 0, 1, -1, -1, 0, -2, -5, -3, -2, 1, -1, -1, 2, 0, 1, -1, 0, 3, 4, 2, 3, 0

Offset: 0

Gus Wiseman, Nov 23 2019

sign

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.

			The sequence of binary expansions together with their runs-resistances and cuts-resistances, and their differences, begins:
   0      (): 0 - 0 =  0
   1     (1): 0 - 1 = -1
   2    (10): 2 - 1 =  1
   3    (11): 1 - 2 = -1
   4   (100): 3 - 2 =  1
   5   (101): 2 - 1 =  1
   6   (110): 3 - 2 =  1
   7   (111): 1 - 3 = -2
   8  (1000): 3 - 3 =  0
   9  (1001): 3 - 2 =  1
  10  (1010): 2 - 1 =  1
  11  (1011): 4 - 2 =  2
  12  (1100): 2 - 2 =  0
  13  (1101): 4 - 2 =  2
  14  (1110): 3 - 3 =  0
  15  (1111): 1 - 4 = -3
  16 (10000): 3 - 4 = -1
  17 (10001): 3 - 3 =  0
  18 (10010): 5 - 2 =  3
  19 (10011): 4 - 2 =  2
  20 (10100): 4 - 2 =  2

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

Positions of 0's are A329865.
Positions of -1's are A329866.
Sorted positions of first appearances are A329868.
Compositions with runs-resistance equal to cuts-resistance are A329864.
Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
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, A329738.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[If[n==0,0,runsres[IntegerDigits[n,2]]-degdep[IntegerDigits[n,2]]],{n,0,100}]

For n > 1, a(2^n) = 3 - n.

For n > 1, a(2^n - 1) = 1 - n.

A382914 Numbers k such that it is not possible to permute a multiset whose multiplicities are the prime indices of k so that the run-lengths are all equal.

Original entry on oeis.org

10, 14, 22, 26, 28, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 66, 68, 69, 74, 76, 78, 82, 85, 86, 87, 88, 92, 93, 94, 95, 102, 104, 106, 111, 114, 115, 116, 118, 119, 122, 123, 124, 129, 130, 134, 136, 138, 141, 142, 145, 146, 148, 152, 153, 155, 156

Offset: 1

Gus Wiseman, Apr 09 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The terms together with their prime indices begin:
  10: {1,3}
  14: {1,4}
  22: {1,5}
  26: {1,6}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  38: {1,8}
  39: {2,6}
  44: {1,1,5}
  46: {1,9}
  51: {2,7}
  52: {1,1,6}
  55: {3,5}
  57: {2,8}
  58: {1,10}
  62: {1,11}
  66: {1,2,5}

For anti-run permutations we have A335126, complement A335127.
Zeros of A382858, anti-run A335125.
For prime indices instead of signature we have A382879, counted by A382915.
For distinct run-lengths we have A382912 (zeros of A382773), complement A382913.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798.
A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.
A304442 counts partitions with equal run-sums, ranks A353833.
A164707 lists numbers whose binary form has equal runs of ones, distinct A328592.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A382857 (firsts A382878), A382771 (firsts A382772).
Cf. A000720, A000961, A001221, A001222, A048767, A181821, A305936, A351013, A382877.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Select[Permutations[nrmptn[#]],SameQ@@Length/@Split[#]&]=={}&]

A385817 Irregular triangle read by rows listing the lengths of maximal runs (sequences of consecutive elements increasing by 1) of binary indices, duplicate rows removed.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 2, 4, 1, 1, 1, 3, 1, 2, 2, 1, 3, 5, 2, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 3, 2, 2, 3, 1, 4, 6, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 1, 5, 1, 2, 1, 2, 1, 2, 2, 4, 2, 1, 1, 3, 3, 3, 2, 4, 1, 5, 7, 2, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 2, 1

Offset: 0

Gus Wiseman, Jul 14 2025

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

This is the triangle A245563, except all duplicates after the first instance of each composition are removed. It lists all compositions in order of their first appearance as a row of A245563.

			The binary indices of 53 are {1,3,5,6}, with maximal runs ((1),(3),(5,6)), with lengths (1,1,2). After removing duplicates, this is our row 16.
Triangle begins:
   0: .
   1: 1
   2: 2
   3: 1 1
   4: 3
   5: 2 1
   6: 1 2
   7: 4
   8: 1 1 1
   9: 3 1
  10: 2 2
  11: 1 3
  12: 5
  13: 2 1 1
  14: 1 2 1
  15: 4 1
  16: 1 1 2
  17: 3 2
  18: 2 3
  19: 1 4
  20: 6
  21: 1 1 1 1

In the following references, "before" is short for "before removing duplicate rows".
Positions of singleton rows appear to be A000071 = A000045-1, before A023758.
Positions of firsts appearances appear to be A001629.
Positions of rows of the form (1,1,...) appear to be A055588 = A001906+1.
First term of each row appears to be A083368.
Row sums appear to be A200648, before A000120.
Row lengths after the first row appear to be A200650+1, before A069010 = A037800+1.
Before the removals we had A245563 (except first term), see A245562, A246029, A328592.
For anti-run ranks we have A385816, before A348366, firsts A052499.
Standard composition numbers of rows are A385818, before A385889.
For anti-runs we have A385886, before A384877, firsts A384878.
Cf. A044813, A048793, A164707, A200649, A242882, A243815, A384175, A384890.

DeleteDuplicates[Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2==#1+1&],{n,0,100}]]

A328607 Numbers whose reversed binary expansion, without the most significant digit, is a necklace.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 26, 28, 30, 31, 32, 48, 52, 56, 58, 60, 62, 63, 64, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 127, 128, 192, 200, 208, 212, 216, 220, 224, 228, 232, 234, 236, 240, 244, 246, 248, 250, 252, 254, 255

Offset: 0

Gus Wiseman, Oct 30 2019

nonn

Offset is 0 to be consistent with A257250.

A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their binary expansions and binary indices begins:
    0:        0 ~ {}
    1:        1 ~ {1}
    2:       10 ~ {2}
    3:       11 ~ {1,2}
    4:      100 ~ {3}
    6:      110 ~ {2,3}
    7:      111 ~ {1,2,3}
    8:     1000 ~ {4}
   12:     1100 ~ {3,4}
   14:     1110 ~ {2,3,4}
   15:     1111 ~ {1,2,3,4}
   16:    10000 ~ {5}
   24:    11000 ~ {4,5}
   26:    11010 ~ {2,4,5}
   28:    11100 ~ {3,4,5}
   30:    11110 ~ {2,3,4,5}
   31:    11111 ~ {1,2,3,4,5}
   32:   100000 ~ {6}
   48:   110000 ~ {5,6}
   52:   110100 ~ {3,5,6}

The dual non-reversed version is A257250.
The dual non-reversed version involving all digits is A065609.
The version involving all digits is A328595.
The non-reversed version is A328668.
Binary necklaces are A000031.
Cf. A000120, A001037, A003714, A008965, A014081, A121016, A164707, A275692, A328594, A328596.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[0,100],#<=1||neckQ[Reverse[Rest[IntegerDigits[#,2]]]]&]

A382772 Set of positions of first appearances in A382771 (permutations of prime indices with distinct run-lengths).

Original entry on oeis.org

1, 6, 12, 96, 360, 1536, 3456, 5184, 5760, 6144, 7776, 13824, 23040, 24576, 55296, 62208, 92160

Offset: 1

Gus Wiseman, Apr 09 2025

			The permutations for n = 12, 96, 360, 1536:
  (1,1,2)  (1,1,1,1,1,2)  (1,1,1,2,2,3)  (1,1,1,1,1,1,1,1,1,2)
  (2,1,1)  (1,1,1,2,1,1)  (1,1,1,3,2,2)  (1,1,1,1,1,1,1,2,1,1)
           (1,1,2,1,1,1)  (2,2,1,1,1,3)  (1,1,1,1,1,1,2,1,1,1)
           (2,1,1,1,1,1)  (2,2,3,1,1,1)  (1,1,1,1,1,2,1,1,1,1)
                          (3,1,1,1,2,2)  (1,1,1,1,2,1,1,1,1,1)
                          (3,2,2,1,1,1)  (1,1,1,2,1,1,1,1,1,1)
                                         (1,1,2,1,1,1,1,1,1,1)
                                         (2,1,1,1,1,1,1,1,1,1)

Positions of first appearances in A382771, by signature A382773.
For equal run-lengths we have A382878, firsts of A382857, zeros A382879.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A328592 lists numbers whose binary form has distinct runs of ones, equal A164707.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A001221, A001222, A003242, A048767, A130091, A238130, A305936, A351013, A351202, A382876.

y=Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,0,100000}];
fip[y_]:=Select[Range[Length[y]],!MemberQ[Take[y,#-1],y[[#]]]&];
fip[Rest[y]]

A329866 Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.

Original entry on oeis.org

1, 3, 16, 30, 33, 48, 55, 56, 59, 60, 67, 68, 72, 79, 95, 97, 110, 112, 118, 120, 121, 125, 134, 135, 137, 143, 145, 158, 160, 195, 196, 219, 220, 225, 231, 241, 250, 258, 270, 280, 286, 291, 292, 315, 316, 351, 381, 382, 390, 391, 393, 399, 415, 416, 431, 432

Offset: 1

Gus Wiseman, Nov 23 2019

nonn

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.

			The sequence of terms together with their binary expansions begins:
    1:         1
    3:        11
   16:     10000
   30:     11110
   33:    100001
   48:    110000
   55:    110111
   56:    111000
   59:    111011
   60:    111100
   67:   1000011
   68:   1000100
   72:   1001000
   79:   1001111
   95:   1011111
   97:   1100001
  110:   1101110
  112:   1110000
  118:   1110110
  120:   1111000
For example, 79 has runs-resistance 3 because we have (1001111) -> (124) -> (111) -> (3), while the cuts-resistance is 4 because we have (1001111) -> (0111) -> (11) -> (1) -> (), so 79 is in the sequence.

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

Positions of -1's in A329867.
The version for runs-resistance equal to cuts-resistance is A329865.
Compositions with runs-resistance equal to cuts-resistance are A329864.
Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
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, A329738, A329868.

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

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.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 23 2019

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.

			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

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

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.
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, A329738, A329865, A329866.

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]}]

A382774 Number of ways to permute the prime indices of n! so that the run-lengths are all different.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 6, 0, 0, 0, 96, 0

Offset: 0

Gus Wiseman, Apr 09 2025

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The prime indices of 24 are {1,1,1,2}, with permutations (1,1,1,2) and (2,1,1,1), so a(4) = 2.

For anti-run permutations we have A335407, see also A335125, A382858.
This is the restriction of A382771 to the factorials A000142, equal A382857.
A022559 counts prime indices of n!, sum A081401.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A328592 lists numbers whose binary form has distinct runs of ones, equal A164707.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A001221, A001222, A003242, A048767, A130091, A181821, A351013, A351202, A360015, A382773, A382879.

Table[Length[Select[Permutations[prix[n!]],UnsameQ@@Length/@Split[#]&]],{n,0,6}]

a(n) = A382771(n!).

A164709 A positive integer n is included if all runs of 1's in binary n are of the same length, and if there are at least two runs of 1's.

Original entry on oeis.org

5, 9, 10, 17, 18, 20, 21, 27, 33, 34, 36, 37, 40, 41, 42, 51, 54, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 99, 102, 108, 119, 129, 130, 132, 133, 136, 137, 138, 144, 145, 146, 148, 149, 160, 161, 162, 164, 165, 168, 169, 170, 195, 198, 204, 216, 219, 231, 238

Offset: 1

Leroy Quet, Aug 23 2009

Clarification: A binary number consists of "runs" completely of 1's alternating with runs completely of 0's. No two or more runs all of the same digit are adjacent.

For the terms of this sequence together with those positive integers that, when written in binary, each contain only one run of 1's, see A164707.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A164707.

r1slQ[n_]:=Module[{idn=Select[Split[IntegerDigits[n,2]],MemberQ[ #,1]&]}, Length[ idn]>1 && Length[Union[Length/@idn]]==1]; Select[ Range[ 250], r1slQ] (* Harvey P. Dale, Sep 29 2018 *)

Extended by Ray Chandler, Mar 15 2010

A257739 Numbers n for which A256999(n) > n; numbers that can be made larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

Original entry on oeis.org

5, 9, 10, 11, 13, 17, 18, 19, 20, 21, 22, 23, 25, 27, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111

Offset: 1

Antti Karttunen, May 18 2015

Note that A256999(a(n)) is always in A257250.

If we define a co-necklace to be a finite sequence that is lexicographically maximal (not minimal) among all of its cyclic rotations, these are numbers whose binary expansion, without the most significant digit, is not a co-necklace. Numbers whose binary expansion, without the most significant digit, is not a necklace are A329367. - Gus Wiseman, Nov 14 2019

			For n = 5 with binary representation "101" if we rotate other bits than the most significant bit (that is, only the two rightmost digits "01") one step to either direction we get "110" = 6 > 5, so 5 can be made larger by such rotations and thus 5 is included in this sequence.
For n = 6 with binary representation "110" no such rotation will yield a larger number and thus 6 is NOT included in this sequence.
For n = 10 with binary representation "1010" if we rotate other bits than the most significant bit (that is, only the three rightmost digits "010") either one step to the left or two steps to the right we get "1100" = 12 > 10, thus 10 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A257250.
Cf. A080541, A080542, A256999.
Numbers whose binary expansion is a necklace are A275692.
Numbers whose binary expansion is a co-necklace are A065609.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose non-msb expansion is a co-necklace are A257250.
Numbers whose non-msb expansion is a necklace are A328668.
Numbers whose reversed non-msb expansion is a necklace are A328607.
Numbers whose non-msb expansion is not a necklace are A329367.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000120, A001037, A003714, A014081, A121016, A164707, A328594, A328596.

reckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[2,100],!reckQ[Rest[IntegerDigits[#,2]]]&] (* Gus Wiseman, Nov 14 2019 *)

cp's OEIS Frontend

A329867 Runs-resistance minus cuts-resistance of the binary expansion of n.

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A382914 Numbers k such that it is not possible to permute a multiset whose multiplicities are the prime indices of k so that the run-lengths are all equal.

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A385817 Irregular triangle read by rows listing the lengths of maximal runs (sequences of consecutive elements increasing by 1) of binary indices, duplicate rows removed.

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A328607 Numbers whose reversed binary expansion, without the most significant digit, is a necklace.

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A382772 Set of positions of first appearances in A382771 (permutations of prime indices with distinct run-lengths).

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A329866 Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.

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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.

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A382774 Number of ways to permute the prime indices of n! so that the run-lengths are all different.

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A164709 A positive integer n is included if all runs of 1's in binary n are of the same length, and if there are at least two runs of 1's.

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