A336231 - OEIS

A336231

Integers whose binary digit expansion has an even number of 0’s between any two consecutive 1’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, 50, 51, 56, 57, 60, 62, 63, 64, 66, 67, 72, 73, 76, 78, 79, 96, 97, 100, 102, 103, 112, 114, 115, 120, 121, 124, 126, 127, 128, 129, 132, 134, 135, 144, 146, 147, 152

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OFFSET

1,3

COMMENTS

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

If m is an odd term and k is odd then 2^k*m+1 is a term. - Robert Israel, Jul 16 2020

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Daniel Glasscock, Joel Moreira, and Florian K. Richter, Additive transversality of fractal sets in the reals and the integers, arXiv:2007.05480 [math.NT], 2020. See Aeven p. 34.

EXAMPLE

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

MAPLE

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

for d from 2 while count < 200 do

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

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

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

od:

[seq(op(sort(convert(S[t], list))), t=0..d-1)]; # Robert Israel, Jul 16 2020

PROG

(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); }