A214415 Numbers n such that prevprime(2^n) AND nextprime(2^n) = 1, where AND is the bitwise AND operator.

2, 4, 6, 8, 12, 15, 16, 23, 25, 30, 37, 53, 55, 57, 67, 75, 76, 81, 82, 84, 95, 108, 129, 132, 135, 139, 143, 155, 160, 163, 180, 181, 188, 192, 203, 204, 210, 222, 244, 263, 273, 277, 280, 287, 289, 295, 297, 308, 315, 319, 325, 330, 341, 367, 370, 393, 394, 406

Offset: 0

Author

Alex Ratushnyak, Aug 07 2012

Keywords

• nonn
• base

Comments

A007053(a(n)) are indices of 1's in A175330. That is, A175330(A007053(a(n)))=1.

Conjecture: the sequence is infinite.

Examples

			4 is in the sequence because (prevprime(2^4) AND nextprime(2^4)) = 13 AND 17 = 1.
		

Crossrefs

Cf. A007053, A175330.

Programs

Java

import java.math.BigInteger;
public class A214415 {
  public static void main (String[] args) {
    BigInteger b1 = BigInteger.valueOf(1);
    BigInteger b2 = BigInteger.valueOf(2);
    for (int n=2; ; n++) {
      BigInteger pwr = b1.shiftLeft(n);
      BigInteger pm  = pwr.subtract(b1);
      BigInteger pp  = pwr.add(b1);
      while (true) {
        if (pm.isProbablePrime(2)) {
            if (pm.isProbablePrime(80)) break;
        }
        pm  = pm.subtract(b2);
      }
      while (true) {
        if (pp.isProbablePrime(2)) {
            if (pp.isProbablePrime(80)) break;
        }
        pp  = pp.add(b2);
      }
      if (pm.and(pp).equals(b1)) {
        System.out.printf("%d, ",n);
      }
    }
  }
}

Mathematica

ba1Q[n_]:=Module[{c=2^n},BitAnd[NextPrime[c],NextPrime[c,-1]]==1]; Select[ Range[ 450],ba1Q] (* Harvey P. Dale, Dec 25 2012 *)

PARI

{ for (n=2,1000,  N = 2^n;
    p1 = precprime(N-1);
    p2 = nextprime(N+1);
    ba = bitand(p1, p2);
    if ( bitand( ba, ba-1 ) == 0, print1(n,", "));
); }
/* Joerg Arndt, Aug 16 2012 */

Python

from itertools import islice
from sympy import prevprime, nextprime
def A214415_gen(): # generator of terms
    n, m = 2, 4
    while True:
        if prevprime(m)&nextprime(m) == 1:
            yield n
        n += 1
        m *= 2
A214415_list = list(islice(A214415_gen(),20)) # Chai Wah Wu, Oct 16 2023