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A339247
The primes that yield twice a prime when each bit of their binary expansion is inverted.
2
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
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
KEYWORD
nonn,base
AUTHOR
Bob Andriesse, Nov 28 2020
STATUS
approved