A065047 Primes which when written in base 2 and prepended with a 1 produce a prime.

3, 5, 13, 29, 37, 43, 71, 83, 101, 113, 163, 193, 211, 223, 257, 311, 317, 347, 479, 509, 547, 577, 613, 643, 673, 709, 787, 823, 853, 877, 883, 907, 1031, 1061, 1181, 1223, 1259, 1283, 1409, 1451, 1481, 1493, 1499, 1511, 1523, 1559, 1583, 1721, 1871, 1973

Offset: 1

Author

Robert G. Wilson v, Nov 05 2001

Keywords

• base
• easy
• nonn

Comments

Primes p such that p and 2^k + p (where k is the smallest power of 2 such that 2^k > p) are primes. - Davide Rotondo, Nov 06 2024

Examples

			13 is in the sequence because 13_10 = 1101_2 and prepending a 1 gives 11101_2 = 29_10, which is a prime.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A004755, A059459.

Programs

Mathematica

Do[p = Prime[n]; d = IntegerDigits[p, 2]; If[ PrimeQ[ FromDigits[ Prepend[d, 1], 2]], Print[p]], {n, 1, 350} ]
Select[Prime[Range[300]],PrimeQ[FromDigits[Join[{1},IntegerDigits[#,2]],2]]&] (* Harvey P. Dale, Apr 10 2023 *)

PARI

{ n=0; t=log(2); for (m=1, 10^9, p=prime(m); if (isprime(p + 2^(1 + log(p)\t)), write("b065047.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 05 2009