A319302 Integers whose binary representation contains a consecutive string of zeros of prime length.

4, 8, 9, 12, 17, 18, 19, 20, 24, 25, 28, 32, 34, 35, 36, 37, 38, 39, 40, 41, 44, 49, 50, 51, 52, 56, 57, 60, 65, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 88, 89, 92, 96, 98, 99, 100, 101, 102, 103, 104, 105, 108, 113, 114, 115, 116, 120

Offset: 1

Author

W. Zane Billings, Sep 16 2018

Keywords

• nonn
• base
• easy

Examples

			81 = (1010001)_2 is a term because it contains a run of zeros of length 3, and 3 is a prime. 16 = (10000)_2 is not a term because it contains only a run of 4 zeros and 4 is not a prime.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004753, A318940.

Programs

Mathematica

Select[Range[120], AnyTrue[ Differences@ Flatten@ Position[ IntegerDigits[ 2*# + 1, 2], 1] - 1, PrimeQ] &] (* Giovanni Resta, Sep 17 2018 *)

PARI

is(n) = my(b=binary(n), i=0); for(k=1, #b, if(b[k]==0, i++); if(b[k]==1 || k==#b, if(ispseudoprime(i), return(1), i=0))); 0 \\ Felix Fröhlich, Sep 17 2018

Python

from re import split
from sympy import isprime
A319302_list, n  = [], 1
while len(A319302_list) < 10000:
    for d in split('1+',bin(n)[2:]):
        if isprime(len(d)):
            A319302_list.append(n)
            break
    n += 1 # Chai Wah Wu, Oct 02 2018

Extensions

More terms from Giovanni Resta, Sep 17 2018