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Numbers whose binary expansion does not have all distinct runs.
4

%I #15 Mar 14 2022 09:28:28

%S 5,9,10,17,18,20,21,22,26,27,33,34,36,37,40,41,42,43,45,46,51,53,54,

%T 58,65,66,68,69,72,73,74,75,76,77,80,81,82,83,84,85,86,87,89,90,91,93,

%U 94,99,100,101,102,105,106,107,108,109,110,117,118,119,122,129

%N Numbers whose binary expansion does not have all distinct runs.

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>

%e The terms together with their binary expansions begin:

%e 5: 101 41: 101001 74: 1001010

%e 9: 1001 42: 101010 75: 1001011

%e 10: 1010 43: 101011 76: 1001100

%e 17: 10001 45: 101101 77: 1001101

%e 18: 10010 46: 101110 80: 1010000

%e 20: 10100 51: 110011 81: 1010001

%e 21: 10101 53: 110101 82: 1010010

%e 22: 10110 54: 110110 83: 1010011

%e 26: 11010 58: 111010 84: 1010100

%e 27: 11011 65: 1000001 85: 1010101

%e 33: 100001 66: 1000010 86: 1010110

%e 34: 100010 68: 1000100 87: 1010111

%e 36: 100100 69: 1000101 89: 1011001

%e 37: 100101 72: 1001000 90: 1011010

%e 40: 101000 73: 1001001 91: 1011011

%e 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 q:= proc(n) uses ListTools; (l-> is(nops(l)<>add(

%p nops(i), i={Split(`=`, l, 1)}) +add(

%p nops(i), i={Split(`=`, l, 0)})))(Bits[Split](n))

%p end:

%p select(q, [$1..200])[]; # _Alois P. Heinz_, Mar 14 2022

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

%o (Python)

%o from itertools import groupby, product

%o def ok(n):

%o runs = [(k, len(list(g))) for k, g in groupby(bin(n)[2:])]

%o return len(runs) > len(set(runs))

%o print([k for k in range(130) if ok(k)]) # _Michael S. Branicky_, Feb 09 2022

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

%Y The complement is A175413, for run-lengths A044813.

%Y The version for standard compositions is A351291, complement A351290.

%Y A000120 counts binary weight.

%Y A011782 counts integer compositions.

%Y A242882 counts compositions with distinct multiplicities.

%Y A318928 gives runs-resistance of binary expansion.

%Y A325545 counts compositions with distinct differences.

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

%Y A334028 counts distinct parts in standard compositions.

%Y A351014 counts distinct runs in standard compositions.

%Y Counting words with all distinct runs:

%Y - A351013 = compositions, for run-lengths A329739.

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

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

%Y - A351200 = patterns, for run-lengths A351292.

%Y - A351202 = permutations of prime factors.

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

%K nonn,base

%O 1,1

%A _Gus Wiseman_, Feb 07 2022