A351205 Numbers whose binary expansion does not have all distinct runs.

5, 9, 10, 17, 18, 20, 21, 22, 26, 27, 33, 34, 36, 37, 40, 41, 42, 43, 45, 46, 51, 53, 54, 58, 65, 66, 68, 69, 72, 73, 74, 75, 76, 77, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 99, 100, 101, 102, 105, 106, 107, 108, 109, 110, 117, 118, 119, 122, 129

Offset: 1

Author

Gus Wiseman, Feb 07 2022

Keywords

• nonn
• base

Examples

			The terms together with their binary expansions begin:
      5:     101     41:  101001     74: 1001010
      9:    1001     42:  101010     75: 1001011
     10:    1010     43:  101011     76: 1001100
     17:   10001     45:  101101     77: 1001101
     18:   10010     46:  101110     80: 1010000
     20:   10100     51:  110011     81: 1010001
     21:   10101     53:  110101     82: 1010010
     22:   10110     54:  110110     83: 1010011
     26:   11010     58:  111010     84: 1010100
     27:   11011     65: 1000001     85: 1010101
     33:  100001     66: 1000010     86: 1010110
     34:  100010     68: 1000100     87: 1010111
     36:  100100     69: 1000101     89: 1011001
     37:  100101     72: 1001000     90: 1011010
     40:  101000     73: 1001001     91: 1011011
For example, 77 has binary expansion 1001101, with runs 1, 00, 11, 0, 1, which are not all distinct, so 77 is in the sequence.
		

Links

• Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Crossrefs

Runs in binary expansion are counted by A005811, distinct A297770.

The complement is A175413, for run-lengths A044813.

The version for standard compositions is A351291, complement A351290.

A000120 counts binary weight.

A011782 counts integer compositions.

A242882 counts compositions with distinct multiplicities.

A318928 gives runs-resistance of binary expansion.

A325545 counts compositions with distinct differences.

A333489 ranks anti-runs, complement A348612, counted by A003242.

A334028 counts distinct parts in standard compositions.

A351014 counts distinct runs in standard compositions.

Counting words with all distinct runs:

- A351013 = compositions, for run-lengths A329739.

- A351016 = binary words, for run-lengths A351017.

- A351018 = binary expansions, for run-lengths A032020.

- A351200 = patterns, for run-lengths A351292.

- A351202 = permutations of prime factors.

Cf. A070939, A085207, A098859, A233564, A238130 or A238279, A283353, A328592, A350952, A351015, A351203.

Programs

Maple

q:= proc(n) uses ListTools; (l-> is(nops(l)<>add(
      nops(i), i={Split(`=`, l, 1)}) +add(
      nops(i), i={Split(`=`, l, 0)})))(Bits[Split](n))
    end:
select(q, [$1..200])[];  # Alois P. Heinz, Mar 14 2022

Mathematica

Select[Range[0,100],!UnsameQ@@Split[IntegerDigits[#,2]]&]

Python

from itertools import groupby, product
def ok(n):
    runs = [(k, len(list(g))) for k, g in groupby(bin(n)[2:])]
    return len(runs) > len(set(runs))
print([k for k in range(130) if ok(k)]) # Michael S. Branicky, Feb 09 2022