A319302 -id:A319302 - cp's OEIS frontend

A318940 Primes whose binary representation contains a consecutive string of zeros of prime length.

17, 19, 37, 41, 71, 73, 79, 83, 89, 101, 103, 113, 131, 137, 139, 149, 151, 157, 163, 167, 179, 193, 197, 199, 211, 227, 229, 233, 241, 257, 263, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 359, 397, 401, 409, 419, 421, 433, 449, 457

Offset: 1

N. J. A. Sloane, Sep 17 2018

Most primes appear to fall into this category. - Charlie Neder, Sep 17 2018

Charlie Neder, Table of n, a(n) for n = 1..949

A subsequence of A319302, which suggested this sequence.

Prime@ Position[Array[If[Length@ # == 0, {0}, Length /@ #] &@ DeleteCases[Split@ IntegerDigits[Prime@ #, 2], ?(First@ # == 1 &)] &, 100], ?(AnyTrue[#, PrimeQ] &), {1}, Heads -> False][[All, 1]] (* Michael De Vlieger, Nov 25 2018 *)
brpQ[p_]:=AnyTrue[Length/@Select[Split[IntegerDigits[p,2]],#[[1]]==0&],PrimeQ]; Select[Prime[Range[100]],brpQ] (* Harvey P. Dale, Jul 30 2025 *)

ok(n)={if(isprime(n), while(n, my(t=valuation(n,2)); if(isprime(t), return(1)); n >>= t + 1)); 0} \\ Andrew Howroyd, Nov 09 2018

More terms from Charlie Neder, Sep 17 2018

A309092 Integers whose hexadecimal representation contains a run of zeros of prime length.

Original entry on oeis.org

256, 512, 768, 1024, 1280, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 3584, 3840, 4096, 4097, 4098, 4099, 4100, 4101, 4102, 4103, 4104, 4105, 4106, 4107, 4108, 4109, 4110, 4111, 4352, 4608, 4864, 5120, 5376, 5632, 5888, 6144, 6400, 6656, 6912, 7168

Offset: 1

W. Zane Billings, Jul 11 2019

			256 = 100_(16) is a term because 100 has a run of two zeros, and two is prime. 258 = 102_(16) is not a term, because its only run of zeros is of length 1, which is not prime.

Cf. A110529, A319302.

Select[Range@ 7168, Select[Split@ IntegerDigits[#, 16], #[[1]] == 0 && PrimeQ@ Length@ # &] != {} &] (* Giovanni Resta, Jul 16 2019 *)

from re import split
from sympy import isprime
seq_list, n = [],1
while len(seq_list) < 10000:
    for d in split('[1-9]+|[a-f]+', format(n,'x')):
        if isprime(len(d)):
            seq_list.append(n)
    n += 1

cp's OEIS Frontend

A318940 Primes whose binary representation contains a consecutive string of zeros of prime length.

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