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

1,3

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

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.

Example

9 is 1001 in binary, with 2 (a prime) consecutive zeroes, 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^(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

Prog

(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); }

Crossrefs

Cf. A007088, A336231.

Sequence in context: A364290 A155562 A048715 * A242662 A335851 A028982

Adjacent sequences: A336229 A336230 A336231 * A336233 A336234 A336235

Keyword

nonn,base,look

Author

Michel Marcus, Jul 13 2020

Status

approved