A292348 "Pri-most" numbers: the majority of bits in the binary representation of these numbers satisfy the following: complementing this bit produces a prime number.

6, 7, 15, 19, 21, 23, 27, 43, 45, 63, 71, 75, 77, 81, 99, 101, 105, 111, 135, 147, 159, 165, 175, 183, 189, 195, 225, 231, 235, 237, 243, 255, 261, 273, 285, 309, 315, 335, 345, 357, 363, 375, 381, 423, 435, 483, 495, 507, 553, 555, 573, 585, 645, 663, 669, 675

Offset: 1

Author

Alex Ratushnyak, Dec 07 2017

Keywords

• nonn
• base

Comments

Conjecture: the sequence is infinite.

Examples

			23 is 10111 in binary, 23 XOR {1,2,4,8,16} = {22,21,19,31,7}, three times a prime was produced, namely 19,31,7, versus two composites, 22 and 21. More primes than composites, therefore 23 is a term.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A137985, A292349.

Programs

Maple

a:= proc(n) option remember; local k; for k from 1+a(n-1) while add(
     `if`(isprime(Bits[Xor](k, 2^i)), 1, -1), i=0..ilog2(k))<1 do od; k
    end: a(0):=0:
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 07 2017

Mathematica

okQ[n_] := Module[{cnt, f}, cnt = Thread[f[n, 2^Range[0, Log[2, n] // Floor]]] /. f -> BitXor // PrimeQ; Count[cnt, True] > Length[cnt]/2];
Select[Range[1000], okQ] (* Jean-François Alcover, Oct 04 2019 *)

Python

from sympy import isprime
for i in range(1000):
  delta = 0  # foundPrime - nonPrime
  bit = 1
  while  bit <= i:
    if isprime(i^bit): delta += 1
    else:              delta -= 1
    bit*=2
  if delta > 0:  print(str(i), end=',')