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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.

%I A292348 #28 May 24 2021 07:33:50
%S A292348 6,7,15,19,21,23,27,43,45,63,71,75,77,81,99,101,105,111,135,147,159,
%T A292348 165,175,183,189,195,225,231,235,237,243,255,261,273,285,309,315,335,
%U A292348 345,357,363,375,381,423,435,483,495,507,553,555,573,585,645,663,669,675
%N 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.
%C A292348 Conjecture: the sequence is infinite.
%H A292348 Alois P. Heinz, <a href="/A292348/b292348.txt">Table of n, a(n) for n = 1..1000</a>
%e A292348 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.
%p A292348 a:= proc(n) option remember; local k; for k from 1+a(n-1) while add(
%p A292348      `if`(isprime(Bits[Xor](k, 2^i)), 1, -1), i=0..ilog2(k))<1 do od; k
%p A292348     end: a(0):=0:
%p A292348 seq(a(n), n=1..100);  # _Alois P. Heinz_, Dec 07 2017
%t A292348 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];
%t A292348 Select[Range[1000], okQ] (* _Jean-François Alcover_, Oct 04 2019 *)
%o A292348 (Python)
%o A292348 from sympy import isprime
%o A292348 for i in range(1000):
%o A292348   delta = 0  # foundPrime - nonPrime
%o A292348   bit = 1
%o A292348   while  bit <= i:
%o A292348     if isprime(i^bit): delta += 1
%o A292348     else:              delta -= 1
%o A292348     bit*=2
%o A292348   if delta > 0:  print(str(i), end=',')
%Y A292348 Cf. A000040, A137985, A292349.
%K A292348 nonn,base
%O A292348 1,1
%A A292348 _Alex Ratushnyak_, Dec 07 2017