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Lesser of twin-bin primes: primes p such that p+2, x and y are primes, where x is concatenation of binary representations of p and p+2, and y is concatenation of binary representations of p+2 and p: x = p * 2^A070939(p+2) + p+2, y = (p+2) * 2^A070939(p) + p.
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%I #23 Jan 26 2014 14:31:17

%S 3,5,269,16649,27689,29129,82889,93239,129629,274199,289169,309479,

%T 336899,349079,371339,374639,415109,454709,463889,492719,1051079,

%U 1063919,1127309,1198289,1209779,1229519,1268789,1350959,1354649,1355279,1392539,1430879,1547129,1551959

%N Lesser of twin-bin primes: primes p such that p+2, x and y are primes, where x is concatenation of binary representations of p and p+2, and y is concatenation of binary representations of p+2 and p: x = p * 2^A070939(p+2) + p+2, y = (p+2) * 2^A070939(p) + p.

%C Conjecture: the sequence is infinite.

%e 269 is in the sequence because the following are three primes: 271, 269 * 512 + 271 = 137999, 271 * 512 + 269 = 139021.

%t Select[Prime[Range[200]], PrimeQ[# + 2] && PrimeQ[FromDigits[Flatten[{IntegerDigits[#, 2], IntegerDigits[# + 2, 2]}], 2]] && PrimeQ[FromDigits[Flatten[{IntegerDigits[# + 2, 2], IntegerDigits[#, 2]}], 2]] &] (* _Alonso del Arte_, Jan 19 2014 *)

%o (Java)

%o import java.math.BigInteger;

%o public class A232239 {

%o public static void main (String[] args) {

%o long bl = 2, next = 3; // bit length, next n such that bl++ for n + 2

%o for (long n = 3; n < 0xffffffffL; n += 2) {

%o long blPrev = bl;

%o if (n == next) { ++bl; next = next * 2 + 1; }

%o if (BigInteger.valueOf(n).isProbablePrime(80) &&

%o BigInteger.valueOf(n + 2).isProbablePrime(80) &&

%o BigInteger.valueOf((n << bl) + n + 2).isProbablePrime(80) &&

%o BigInteger.valueOf(((n + 2) << blPrev) + n).isProbablePrime(80))

%o System.out.printf("%d, ", n);

%o }

%o }

%o }

%Y Cf. A001359, A070939, A232237, A232238.

%K nonn,base,less

%O 1,1

%A _Alex Ratushnyak_, Nov 20 2013