A287117 Numbers with no odd prime binary proper prefixes.

1, 2, 3, 4, 5, 8, 9, 16, 17, 18, 19, 32, 33, 36, 37, 64, 65, 66, 67, 72, 73, 128, 129, 130, 131, 132, 133, 144, 145, 256, 257, 258, 259, 260, 261, 264, 265, 266, 267, 288, 289, 290, 291, 512, 513, 516, 517, 518, 519, 520, 521, 522, 523, 528, 529, 530, 531, 532, 533, 534, 535

Offset: 1

Author

Dan Brumleve, May 20 2017

Keywords

• nonn
• easy
• base

Examples

			131, while prime itself, has proper binary prefixes 65, 32, 16, 8, 4, 2, 1, none of which are odd primes.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..2881
• Dan Brumleve, Does the sum of reciprocals of all prime-prefix-free numbers converge?, Math StackExchange, May 20 2017.

Programs

Mathematica

Select[Range@535, AllTrue[ Floor[#/2 ^ Range@Log2@#], ! (# > 2 && PrimeQ[#]) &] &] (* Giovanni Resta, May 20 2017 *)

Perl

sub isp {
  my $x = shift;
  for my $d (2 .. $x - 1) {
    return 0 if $x % $d == 0;
  }
  return 1;
}
sub rots {
  my $x = shift;
  my @x;
  while ($x > 5) {
    $x = int($x / 2);
    push @x, $x;
  }
  @x
}
for my $i (1 .. $ARGV[0] // 200) {
  my @np = grep isp($_), rots($i);
  push @z, $i if @np == 0;
}
print join(", ", @z) . "\n";