A163498 - OEIS

%I #19 Aug 22 2018 20:44:10

%S 2,3,11,13,23,37,43,59,79,89,103,109,113,139,149,181,193,197,227,239,

%T 263,269,281,283,307,401,433,443,449,457,463,503,523,547,587,617,653,

%U 673,691,811,821,823,829,839,877,887,911,937,967,1021,1049,1061,1063

%N Those primes p that after each is written in binary, and a 0 is inserted before every 1, then the value of this new number is also prime.

%C Equal to primes p such that when written in binary, and a 0 is inserted after every binary digit 1, results in 2 times a prime number. For example, 13 is in the list as 13 is 1101 in binary.  Inserting 0 after every 1 results in 1010010 = 82 decimal, which is 2*41 and 41 is prime. - _Chai Wah Wu_, Jul 29 2014

%H Chai Wah Wu, <a href="/A163498/b163498.txt">Table of n, a(n) for n = 1..2000</a>

%e 13 in binary is 1101. Insert a 0 before every 1, and we have 0101001, which is 41 in decimal (ignoring the leading 0 in the binary representation). Since 41 is also prime, then 13 is included in this sequence.

%t a = {}; For[n = 1, n < 1000, n++, b = IntegerDigits[Prime[n], 2]; c = {}; For[k = 1, k < Length[b] + 1, k++, AppendTo[c, 0]; If[b[[k]] == 1, AppendTo[c, 1]]]; If[PrimeQ[FromDigits[c, 2]], AppendTo[a, Prime[n]]]]; a (* _Stefan Steinerberger_, Aug 05 2009 *)

%t Select[Prime[Range[200]],PrimeQ[FromDigits[Flatten[IntegerDigits[#,2]/.(1-> {0,1})],2]]&] (* _Harvey P. Dale_, Aug 22 2018 *)

%o (Python) from sympy import prime, isprime

%o [prime(n) for n in range(1,1000) if isprime(int(bin(prime(n)).replace('1','01'),2))] # _Chai Wah Wu_, Jul 28 2014

%Y Cf. A163499.

%K base,nonn

%O 1,1

%A _Leroy Quet_, Jul 29 2009

%E More terms from _Stefan Steinerberger_, Aug 05 2009

%E More terms from _Chai Wah Wu_, Jul 28 2014