cp's OEIS Frontend

A287638 Odd primes with no other odd primes as prefixes in binary.

%I A287638 #20 Sep 24 2023 10:23:37
%S A287638 3,5,17,19,37,67,73,131,257,521,523,577,1033,1039,1061,1063,1069,1153,
%T A287638 1163,2053,2069,2081,2083,2089,2113,2129,2131,2137,2141,2143,2333,
%U A287638 4099,4111,4129,4153,4177,4229,4231,4241,4243,4261,4271,4273,4637,4639,4643,4649,4651,4657,4663
%N A287638 Odd primes with no other odd primes as prefixes in binary.
%C A287638 The sum of the reciprocals converges by the Kraft-McMillan inequality.
%C A287638 Odd primes p such that iterating the map A029578 on p reaches 2 without going through a prime. - _Ya-Ping Lu_, Oct 20 2021
%H A287638 Charles R Greathouse IV, <a href="/A287638/b287638.txt">Table of n, a(n) for n = 1..10000</a>
%H A287638 Dan Brumleve, <a href="https://math.stackexchange.com/questions/2288648/does-the-sum-of-reciprocals-of-all-prime-prefix-free-numbers-converge">Does the sum of reciprocals of all prime-prefix-free numbers converge?</a>, Math StackExchange, May 20 2017.
%H A287638 Wikipedia, <a href="https://en.wikipedia.org/wiki/Kraft%E2%80%93McMillan_inequality">Kraft-McMillan inequality</a>
%e A287638 4663 is an odd prime with proper binary prefixes 2331, 1165, 582, 291, 145, 72, 36, 18, 9, 4, 2, 1, and none of these are odd primes.
%t A287638 Select[Prime@ Range[2, 800], Function[w, NoneTrue[Array[FromDigits[w[[1 ;; #]], 2] &, Length[w] - 1], And[PrimeQ[#], # != 2] &]]@ IntegerDigits[#, 2] &] (* _Michael De Vlieger_, Oct 20 2021 *)
%o A287638 (Perl)
%o A287638 sub isp {
%o A287638   my $x = shift;
%o A287638   return 0 if $x < 2;
%o A287638   for my $d (2 .. $x - 1) {
%o A287638     return 0 if $x % $d == 0;
%o A287638   }
%o A287638   return 1;
%o A287638 }
%o A287638 sub rots {
%o A287638   my $x = shift;
%o A287638   my @x;
%o A287638   while ($x > 5) {
%o A287638     $x = int($x / 2);
%o A287638     push @x, $x;
%o A287638   }
%o A287638   @x
%o A287638 }
%o A287638 for my $i (1 .. $ARGV[0] // 8000) {
%o A287638   my @np = grep isp($_), rots($i);
%o A287638   push @z, $i if isp($i) && $i % 2 && @np == 0;
%o A287638 }
%o A287638 print join(", ", @z) . "\n";
%o A287638 (PARI) is(n)=if(!isprime(n) || n<3, return(0)); while(n>3, if(isprime(n>>=1), return(n==2))); 1 \\ _Charles R Greathouse IV_, Jun 12 2017
%Y A287638 Odd prime elements of A287117.
%Y A287638 Cf. A029578.
%K A287638 nonn,easy,base
%O A287638 1,1
%A A287638 _Dan Brumleve_, May 28 2017