A065047 - OEIS

A065047

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

%I #18 Nov 19 2024 01:05:48

%S 3,5,13,29,37,43,71,83,101,113,163,193,211,223,257,311,317,347,479,

%T 509,547,577,613,643,673,709,787,823,853,877,883,907,1031,1061,1181,

%U 1223,1259,1283,1409,1451,1481,1493,1499,1511,1523,1559,1583,1721,1871,1973

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

%C 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

%H Harry J. Smith, <a href="/A065047/b065047.txt">Table of n, a(n) for n = 1..1000</a>

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

%t Do[p = Prime[n]; d = IntegerDigits[p, 2]; If[ PrimeQ[ FromDigits[ Prepend[d, 1], 2]], Print[p]], {n, 1, 350} ]

%t Select[Prime[Range[300]],PrimeQ[FromDigits[Join[{1},IntegerDigits[#,2]],2]]&] (* _Harvey P. Dale_, Apr 10 2023 *)

%o (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

%Y Cf. A004755, A059459.

%K base,easy,nonn

%O 1,1

%A _Robert G. Wilson v_, Nov 05 2001