A166484 Prime sums of three Fermat numbers: primes of form 2^2^x + 2^2^y + 5.

11, 13, 23, 37, 263, 277, 65543, 65557, 4295032837

Offset: 1

Author

Jonathan Vos Post, Oct 14 2009, Oct 22 2009

Keywords

• hard
• nonn

Comments

One can have a prime sum of two Fermat Primes, starting with 2 + 3 = 5.

Hence this current sequence is a proper subset of prime sums of a Fermat prime number of Fermat numbers, which in turn is a proper subset of prime sums of a Fermat number of Fermat numbers.

According to the Maple 9 primality test, the next term is larger than 10^300 if it exists. - R. J. Mathar, Oct 16 2009

At least one of the three Fermat numbers must be 3 because all Fermat numbers greater than 3 are equal to 2 (mod 3). Hence, the sum of three Fermat numbers greater than 3 is always a multiple of 3.

The next term, if it exists, has at least 1262612 digits. - Arkadiusz Wesolowski, Mar 06 2011

Examples

			a(1) = A000215(0) + A000215(0) + A000215(1) = 3 + 3 + 5 = 11, which is prime.
a(2) = A000215(0) + A000215(1) + A000215(1) = 3 + 5 + 5 = 13, which is prime.
a(3) = A000215(0) + A000215(0) + A000215(2) = 3 + 3 + 17 = 23, which is prime.
a(4) = A000215(0) + A000215(2) + A000215(2) = 3 + 17 + 17 = 37, which is prime.
		

Links

• G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 65557

Crossrefs

Cf. A000040, A155877, A019434.

Programs

PARI

for(x=1,9,for(y=1,x,if(isprime(t=2^2^x+2^2^y+5), print1(t", ")))) \\ Charles R Greathouse IV, Apr 29 2016

Formula

A155877 INTERSECTION A000040.

{p = (2^(2^a) + 1) + (2^(2^b) + 1) + (2^(2^c) + 1) for nonnegative integers a, b, c, such that p is prime}.

Extensions

a(9) from R. J. Mathar, Oct 16 2009

Definition improved by Arkadiusz Wesolowski, Feb 16 2011