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A336232 Integers whose binary digit expansion has a prime number of 0’s between any two consecutive 1’s. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A036349 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

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Last modified December 5 10:11 EST 2021. Contains 349543 sequences. (Running on oeis4.)