A336231 - OEIS

%I #17 Sep 22 2024 17:33:54

%S 0,1,2,3,4,6,7,8,9,12,14,15,16,18,19,24,25,28,30,31,32,33,36,38,39,48,

%T 50,51,56,57,60,62,63,64,66,67,72,73,76,78,79,96,97,100,102,103,112,

%U 114,115,120,121,124,126,127,128,129,132,134,135,144,146,147,152

%N Integers whose binary digit expansion has an even number of 0’s between any two consecutive 1’s.

%C If m is a term then 2*m is a term too.

%C If m is an odd term and k is odd then 2^k*m+1 is a term. - _Robert Israel_, Jul 16 2020

%H Robert Israel, <a href="/A336231/b336231.txt">Table of n, a(n) for n = 1..10000</a>

%H Daniel Glasscock, Joel Moreira, and Florian K. Richter, <a href="https://arxiv.org/abs/2007.05480">Additive transversality of fractal sets in the reals and the integers</a>, arXiv:2007.05480 [math.NT], 2020. See Aeven p. 34.

%e 9 is 1001 in binary, with 2 (an even number) consecutive zeros, so 9 is a term.

%p B[1]:= {1}: S[0]:= {0}: S[1]:= {1}: count:= 2:

%p for d from 2 while count < 200 do

%p   B[d]:= map(op, {seq(map(t -> t*2^k+1, B[d-k]), k=1..d-1,2)});

%p   S[d]:= B[d] union map(`*`, S[d-1], 2);

%p   count:= count+nops(S[d]);

%p od:

%p [seq(op(sort(convert(S[t], list))), t=0..d-1)]; # _Robert Israel_, Jul 16 2020

%o (PARI) isok(n) = {my(vpos = select(x->(x==1), binary(n), 1)); for (i=1, #vpos-1, if ((vpos[i+1]-vpos[i]-1) % 2, return (0));); return(1);}

%Y Cf. A007088, A060142, A336232.

%K nonn,base

%O 1,3

%A _Michel Marcus_, Jul 13 2020