cp's OEIS Frontend

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

%I A351205 #15 Mar 14 2022 09:28:28
%S A351205 5,9,10,17,18,20,21,22,26,27,33,34,36,37,40,41,42,43,45,46,51,53,54,
%T A351205 58,65,66,68,69,72,73,74,75,76,77,80,81,82,83,84,85,86,87,89,90,91,93,
%U A351205 94,99,100,101,102,105,106,107,108,109,110,117,118,119,122,129
%N A351205 Numbers whose binary expansion does not have all distinct runs.
%H A351205 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A351205 The terms together with their binary expansions begin:
%e A351205       5:     101     41:  101001     74: 1001010
%e A351205       9:    1001     42:  101010     75: 1001011
%e A351205      10:    1010     43:  101011     76: 1001100
%e A351205      17:   10001     45:  101101     77: 1001101
%e A351205      18:   10010     46:  101110     80: 1010000
%e A351205      20:   10100     51:  110011     81: 1010001
%e A351205      21:   10101     53:  110101     82: 1010010
%e A351205      22:   10110     54:  110110     83: 1010011
%e A351205      26:   11010     58:  111010     84: 1010100
%e A351205      27:   11011     65: 1000001     85: 1010101
%e A351205      33:  100001     66: 1000010     86: 1010110
%e A351205      34:  100010     68: 1000100     87: 1010111
%e A351205      36:  100100     69: 1000101     89: 1011001
%e A351205      37:  100101     72: 1001000     90: 1011010
%e A351205      40:  101000     73: 1001001     91: 1011011
%e A351205 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.
%p A351205 q:= proc(n) uses ListTools; (l-> is(nops(l)<>add(
%p A351205       nops(i), i={Split(`=`, l, 1)}) +add(
%p A351205       nops(i), i={Split(`=`, l, 0)})))(Bits[Split](n))
%p A351205     end:
%p A351205 select(q, [$1..200])[];  # _Alois P. Heinz_, Mar 14 2022
%t A351205 Select[Range[0,100],!UnsameQ@@Split[IntegerDigits[#,2]]&]
%o A351205 (Python)
%o A351205 from itertools import groupby, product
%o A351205 def ok(n):
%o A351205     runs = [(k, len(list(g))) for k, g in groupby(bin(n)[2:])]
%o A351205     return len(runs) > len(set(runs))
%o A351205 print([k for k in range(130) if ok(k)]) # _Michael S. Branicky_, Feb 09 2022
%Y A351205 Runs in binary expansion are counted by A005811, distinct A297770.
%Y A351205 The complement is A175413, for run-lengths A044813.
%Y A351205 The version for standard compositions is A351291, complement A351290.
%Y A351205 A000120 counts binary weight.
%Y A351205 A011782 counts integer compositions.
%Y A351205 A242882 counts compositions with distinct multiplicities.
%Y A351205 A318928 gives runs-resistance of binary expansion.
%Y A351205 A325545 counts compositions with distinct differences.
%Y A351205 A333489 ranks anti-runs, complement A348612, counted by A003242.
%Y A351205 A334028 counts distinct parts in standard compositions.
%Y A351205 A351014 counts distinct runs in standard compositions.
%Y A351205 Counting words with all distinct runs:
%Y A351205 - A351013 = compositions, for run-lengths A329739.
%Y A351205 - A351016 = binary words, for run-lengths A351017.
%Y A351205 - A351018 = binary expansions, for run-lengths A032020.
%Y A351205 - A351200 = patterns, for run-lengths A351292.
%Y A351205 - A351202 = permutations of prime factors.
%Y A351205 Cf. A070939, A085207, A098859, A233564, A238130 or A238279, A283353, A328592, A350952, A351015, A351203.
%K A351205 nonn,base
%O A351205 1,1
%A A351205 _Gus Wiseman_, Feb 07 2022