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A336232 Integers whose binary digit expansion has a prime number of 0’s between any two consecutive 1’s.

%I A336232 #21 Jul 18 2020 02:39:59
%S A336232 0,1,2,4,8,9,16,17,18,32,34,36,64,65,68,72,73,128,130,136,137,144,145,
%T A336232 146,256,257,260,272,273,274,288,290,292,512,514,520,521,544,546,548,
%U A336232 576,577,580,584,585,1024,1028,1040,1041,1042,1088,1089,1092,1096,1097
%N A336232 Integers whose binary digit expansion has a prime number of 0’s between any two consecutive 1’s.
%C A336232 If m is a term then 2*m is a term too.
%C A336232 If m is an odd term and p is prime then 2^(p+1)*m+1 is a term. - _Robert Israel_, Jul 15 2020
%H A336232 Robert Israel, <a href="/A336232/b336232.txt">Table of n, a(n) for n = 1..10000</a>
%H A336232 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 Aprime p. 34.
%H A336232 Benjamin Matson and Elizabeth Sattler, <a href="https://arxiv.org/abs/1708.08511">S-limited shifts</a>, arXiv:1708.08511 [math.DS], 2017. See page 2.
%e A336232 9 is 1001 in binary, with 2 (a prime) consecutive zeroes, so 9 is a term.
%p A336232 B[1]:= {1}: S[0]:= {0}: S[1]:= {1}: count:= 2:
%p A336232 for d from 2 while count < 200 do
%p A336232   B[d]:= map(op,{seq(map(t -> t*2^(p+1)+1,B[d-p-1]),p=select(isprime,[$2..d-2]))});
%p A336232   S[d]:= B[d] union map(`*`,S[d-1],2);
%p A336232   count:= count+nops(S[d]);
%p A336232 od:
%p A336232 [seq(op(sort(convert(S[t],list))),t=0..d-1)]; # _Robert Israel_, Jul 16 2020
%o A336232 (PARI) isok(n) = {my(vpos = select(x->(x==1), binary(n), 1)); for (i=1, #vpos-1, if (!isprime(vpos[i+1]-vpos[i]-1), return (0));); return(1);}
%Y A336232 Cf. A007088, A336231.
%K A336232 nonn,base,look
%O A336232 1,3
%A A336232 _Michel Marcus_, Jul 13 2020