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

0, 1, 2, 4, 8, 9, 16, 17, 18, 32, 34, 36, 64, 65, 68, 72, 73, 128, 130, 136, 137, 144, 145, 146, 256, 257, 260, 272, 273, 274, 288, 290, 292, 512, 514, 520, 521, 544, 546, 548, 576, 577, 580, 584, 585, 1024, 1028, 1040, 1041, 1042, 1088, 1089, 1092, 1096, 1097

Offset: 1

Author

Michel Marcus, Jul 13 2020

Keywords

• nonn
• base
• look

Comments

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

If m is an odd term and p is prime then 2^(p+1)*m+1 is a term. - Robert Israel, Jul 15 2020

Examples

			9 is 1001 in binary, with 2 (a prime) consecutive zeroes, so 9 is a term.
		

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 Aprime p. 34.
• Benjamin Matson and Elizabeth Sattler, S-limited shifts, arXiv:1708.08511 [math.DS], 2017. See page 2.

Crossrefs

Cf. A007088, A336231.

Programs

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^(p+1)+1,B[d-p-1]),p=select(isprime,[$2..d-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

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);}