A339268 The smallest prime that becomes 2 * prime(n), when all the bits in its binary expansion are inverted, or -1 if no such prime exists.

11, 549755813881, 53, 17, 41, 37, 16349, 89, 977, 197, 193, 181, 173, 937, 929, 149, 137, 389, 1913, 881, 877, 353, 857, 3917, 317, 821, 3889, 809, 293, 797, 257, 761, 3821, 65257, 3797, 3793, 709, 1721, 3761, 677, 1048217, 661, 641, 3709, 3701, 3697, 601, 577

Offset: 1

Author

Bob Andriesse, Nov 29 2020

Keywords

• nonn
• base

Comments

Conjecture: a(n) > 0 for all n.

a(6705), for prime(6705) = 67289, is either -1 or greater than (2^62500 - 1) * 262144 + 127565, which has 18820 digits. - Michael S. Branicky, Dec 05 2020

Examples

			a(1) = 11, because 11 = 1011_2 -inv-> 0100_2 = 4 = 2 * 2.
a(2) = 549755813881, because 549755813881 = 111111111111111111111111111111111111001_2 -inv-> 110_2 = 6 = 2 * 3. No smaller prime generates 3.
a(3) = 53, because 53 = 110101_2 -inv-> 001010_2 = 10 = 2 * 5.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..6704

Crossrefs

Cf. A035327, A339247.

Programs

Mathematica

a[n_] := Module[{q = 2*Prime[n], m, r}, m = 2^Ceiling@Log2[q]; r = m - q - 1; While[r < q || ! PrimeQ[r], r += m; m *= 2]; r]; Array[a, 48] (* Amiram Eldar, Dec 04 2020 *)

PARI

a(n) = {my(b = apply(x->1-x, binary(2*prime(n))), e=#b, q=fromdigits(b, 2)+2^e); while (!isprime(q), e++; q+=2^e; q); q;} \\ Michel Marcus, Dec 04 2020

Python

from sympy import isprime
from sympy import prime
for i in range(1,50):
  d=2*prime(i)
  l=len(bin(d).lstrip('0b'))
  xor=2**l-1
  p=d^xor+2**l
  while not isprime(p):
   l+=1
   p+=2**l
  print(p,end=', ')