A274000 Primes p of the form k^2 + 1 that divide 4^k - 1.

5, 17, 257, 65537, 148997, 405458497, 3497539601, 139251776898727937

Offset: 1

Author

Jaroslav Krizek, Jun 06 2016

Keywords

• nonn
• more

Comments

Prime terms from A273999.

The first 4 known Fermat primes > 3 from A019434 are in this sequence.

Conjecture: also primes p of the form n^2+1 such that divides (4^k)^n-1 for all k >= 0. Example: 17 = 4^2+1 is a term because divides (4^k)^4-1 for all k>=0: 0/17 = 0 (k=0); 255/17 = 15 (k=1); 65535/17 = 3855 (k=2); 16777215/17 = 986895 (k=3); 4294967295/17 = 252645135 (k=4); 1099511627775/17 = 64677154575 (k=5); ...

a(8) > 10^15. - Jacques Tramu, Sep 17 2018

a(9) > 1.7*10^18. - Giovanni Resta, Nov 12 2018

Examples

			17 = 4^2 + 1 is a term because it divides 4^4 - 1; 255/17 = 15.
		

Crossrefs

Cf. A000215, A002496, A019434, A273999.

Subsequence of A002496 (primes of the form n^2+1).

Supersequence of A274002.

Programs

PARI

is(n) = ceil(sqrt(n-1))==sqrtint(n-1) && Mod(4, n)^(sqrtint(n))==1
for(n=0, 1e12, if(is(n^2+1), if(ispseudoprime(n^2+1), print1(n^2+1, ", ")))) \\ Felix Fröhlich, Jun 12 2016

Extensions

a(8) from Giovanni Resta, Nov 12 2018