A357759 - OEIS

%I #18 Oct 09 2024 18:31:16

%S 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,

%T 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,

%U 49,50,51,52,53,54,55,57,58,59,60,61,62,63,64,65,66,67,68,69,70,72,73,74,75

%N Numbers k such that in the binary expansion of k, the Hamming weight of each block differs by at most 2 from every other block of the same length.

%C Leading zeros in binary expansions are ignored.

%C For any n > 0, there are A274005(n)/2 positive terms with binary length n.

%C Empirically, if t is a term, then at least one of 2*t or 2*t + 1 is also a term.

%C If t is a term, then floor(t/2) is also a term.

%C The complement with respect to the nonnegative integers is 56, 71, 112, 113, 120, 135, 142, 143, 176, 184, 199, ... - _Andrew Howroyd_, Oct 09 2024

%H Andrew Howroyd, <a href="/A357759/b357759.txt">Table of n, a(n) for n = 1..7658</a>

%H Rémy Sigrist, <a href="/A357759/a357759.gp.txt">PARI program</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%e For k = 12:

%e - the binary expansion of k is "1100",

%e - blocks of length 1 have Hamming weight 0 or 1,

%e - blocks of length 2 have Hamming weight 0, 1 or 2,

%e - blocks of length 3 have Hamming weight 1 or 2,

%e - blocks of length 4 have Hamming weight 2,

%e - so 12 belongs to the sequence.

%e For k = 56:

%e - the binary expansion of 44 is "111000",

%e - blocks of length 3 have Hamming weight 0, 1, 2 or 3,

%e - so 56 does not belong to the sequence.

%o (PARI) \\ See Links section.

%o (Python)

%o def ok(n):

%o     b = bin(n)[2:]

%o     if "000" in b and "111" in b: return False

%o     for l in range(4, len(b)-1):

%o         h = set(b[i:i+l].count("1") for i in range(len(b)-l+1))

%o         if max(h) - min(h) > 2: return False

%o     return True

%o print([k for k in range(69) if ok(k)]) # _Michael S. Branicky_, Oct 12 2022

%Y Cf. A274005, A357758.

%K nonn,base

%O 1,3

%A _Rémy Sigrist_, Oct 12 2022

%E a(69) onwards from _Andrew Howroyd_, Oct 09 2024