A339247 The primes that yield twice a prime when each bit of their binary expansion is inverted.

11, 17, 37, 41, 53, 59, 89, 101, 113, 137, 149, 173, 181, 193, 197, 229, 233, 241, 251, 257, 293, 317, 353, 389, 449, 521, 541, 557, 569, 577, 601, 641, 661, 677, 709, 761, 769, 797, 809, 821, 829, 857, 877, 881, 929, 937

Offset

1,1

Comments

Primes p such that A035327(p)/2 is prime.

Obviously the resulting prime is smaller than the starting prime.

Conjecture: Each of the primes can be created by applying this transformation T(p) to at least one other prime. T(11)=2, T(549755813881)=3, T(53)=5, T(17)=7, T(41)=11.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(1) = 11 = 1011_2 —inv-> 100_2 = 4 = 2 * 2.
a(2) = 17 = 10001_2 -inv-> 1110_2 = 14 = 2 * 7.
a(3) = 37 = 100101_2 -inv-> 11010_2 = 26 = 2 * 13.

Maple

filter:= proc(n) isprime(n) and isprime((2^(1+ilog2(n))-1-n)/2) end proc:
select(filter, [seq(i, i=3..1000, 2)]); # Robert Israel, Jun 27 2023

Mathematica

Select[Range[1000], And @@ PrimeQ[{#, (2^Ceiling @ Log2[#] - # - 1)/2}] &] (* Amiram Eldar, Dec 01 2020 *)

Prog

(Python)
from sympy import isprime
from sympy import prime
for i in range(1, 201):
  j=prime(i)
  xor=2**len(bin(j).strip('0b'))-1
  p=(j^xor)//2
  if isprime(p):
   print(j, end=', ')
(PARI) isok(p) = if ((p>2) && isprime(p), isprime(fromdigits(apply(x->1-x, binary(p)), 2)/2)); \\ Michel Marcus, Dec 01 2020

Crossrefs

Cf. A000040, A035327, A339268.

Sequence in context: A098797 A271999 A317453 * A317678 A098649 A078872

Adjacent sequences: A339244 A339245 A339246 * A339248 A339249 A339250

Keyword

nonn,base

Author

Bob Andriesse, Nov 28 2020

Status

approved