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

1,1

Comments

Conjecture: the sequence is infinite.

Links

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

Example

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.

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 *)

Prog

(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=', ')

Crossrefs

Cf. A000040, A137985, A292349.

Sequence in context: A315842 A085512 A165767 * A319185 A196223 A106680

Adjacent sequences: A292345 A292346 A292347 * A292349 A292350 A292351

Keyword

nonn,base

Author

Alex Ratushnyak, Dec 07 2017

Status

approved