login
Integers whose binary digit expansion has a prime number of 0’s between any two consecutive 1’s.
2

%I #21 Jul 18 2020 02:39:59

%S 0,1,2,4,8,9,16,17,18,32,34,36,64,65,68,72,73,128,130,136,137,144,145,

%T 146,256,257,260,272,273,274,288,290,292,512,514,520,521,544,546,548,

%U 576,577,580,584,585,1024,1028,1040,1041,1042,1088,1089,1092,1096,1097

%N Integers whose binary digit expansion has a prime 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 p is prime then 2^(p+1)*m+1 is a term. - _Robert Israel_, Jul 15 2020

%H Robert Israel, <a href="/A336232/b336232.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 Aprime p. 34.

%H 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 9 is 1001 in binary, with 2 (a prime) consecutive zeroes, 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^(p+1)+1,B[d-p-1]),p=select(isprime,[$2..d-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 (!isprime(vpos[i+1]-vpos[i]-1), return (0));); return(1);}

%Y Cf. A007088, A336231.

%K nonn,base,look

%O 1,3

%A _Michel Marcus_, Jul 13 2020