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

%I A214415 #32 Oct 16 2023 12:41:19
%S A214415 2,4,6,8,12,15,16,23,25,30,37,53,55,57,67,75,76,81,82,84,95,108,129,
%T A214415 132,135,139,143,155,160,163,180,181,188,192,203,204,210,222,244,263,
%U A214415 273,277,280,287,289,295,297,308,315,319,325,330,341,367,370,393,394,406
%N A214415 Numbers n such that prevprime(2^n) AND nextprime(2^n) = 1, where AND is the bitwise AND operator.
%C A214415 A007053(a(n)) are indices of 1's in A175330. That is, A175330(A007053(a(n)))=1.
%C A214415 Conjecture: the sequence is infinite.
%e A214415 4 is in the sequence because (prevprime(2^4) AND nextprime(2^4)) = 13 AND 17 = 1.
%t A214415 ba1Q[n_]:=Module[{c=2^n},BitAnd[NextPrime[c],NextPrime[c,-1]]==1]; Select[ Range[ 450],ba1Q] (* _Harvey P. Dale_, Dec 25 2012 *)
%o A214415 (Java)
%o A214415 import java.math.BigInteger;
%o A214415 public class A214415 {
%o A214415   public static void main (String[] args) {
%o A214415     BigInteger b1 = BigInteger.valueOf(1);
%o A214415     BigInteger b2 = BigInteger.valueOf(2);
%o A214415     for (int n=2; ; n++) {
%o A214415       BigInteger pwr = b1.shiftLeft(n);
%o A214415       BigInteger pm  = pwr.subtract(b1);
%o A214415       BigInteger pp  = pwr.add(b1);
%o A214415       while (true) {
%o A214415         if (pm.isProbablePrime(2)) {
%o A214415             if (pm.isProbablePrime(80)) break;
%o A214415         }
%o A214415         pm  = pm.subtract(b2);
%o A214415       }
%o A214415       while (true) {
%o A214415         if (pp.isProbablePrime(2)) {
%o A214415             if (pp.isProbablePrime(80)) break;
%o A214415         }
%o A214415         pp  = pp.add(b2);
%o A214415       }
%o A214415       if (pm.and(pp).equals(b1)) {
%o A214415         System.out.printf("%d, ",n);
%o A214415       }
%o A214415     }
%o A214415   }
%o A214415 }
%o A214415 (PARI)
%o A214415 { for (n=2,1000,  N = 2^n;
%o A214415     p1 = precprime(N-1);
%o A214415     p2 = nextprime(N+1);
%o A214415     ba = bitand(p1, p2);
%o A214415     if ( bitand( ba, ba-1 ) == 0, print1(n,", "));
%o A214415 ); }
%o A214415 /* _Joerg Arndt_, Aug 16 2012 */
%o A214415 (Python)
%o A214415 from itertools import islice
%o A214415 from sympy import prevprime, nextprime
%o A214415 def A214415_gen(): # generator of terms
%o A214415     n, m = 2, 4
%o A214415     while True:
%o A214415         if prevprime(m)&nextprime(m) == 1:
%o A214415             yield n
%o A214415         n += 1
%o A214415         m *= 2
%o A214415 A214415_list = list(islice(A214415_gen(),20)) # _Chai Wah Wu_, Oct 16 2023
%Y A214415 Cf. A007053, A175330.
%K A214415 nonn,base
%O A214415 0,1
%A A214415 _Alex Ratushnyak_, Aug 07 2012