A333803 Primes p such that 2*p+1 and 4*p^2+1 are also prime.

2, 3, 5, 233, 653, 683, 1013, 1973, 2003, 2393, 2543, 2753, 3023, 3413, 5003, 5333, 7043, 7823, 8663, 9293, 10613, 13463, 13913, 14783, 15233, 15923, 16823, 18233, 20693, 20753, 21713, 21803, 22433, 27743, 27983, 29723, 30323, 30773, 31253, 31793, 32003, 33053, 33623, 33773, 34283, 36083, 37013

Offset

1,1

Comments

The generalized Bunyakovsky conjecture implies there are infinitely many terms.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Mathematics StackExchange,Find all p such that p, 2p+1, 4p^2+1 are prime

Example

a(3)=5 is a member because 5, 2*5+1=11 and 4*5^2+1= 101 are all prime.

Maple

filter:= proc(n)
  isprime(n) and isprime(2*n+1) and isprime(4*n^2+1)
end proc:
select(filter, [2, 3, seq(i, i=5..10^5, 6)]);

Mathematica

Select[Prime[Range[4000]], AllTrue[{2#+1, 4#^2+1}, PrimeQ]&] (* Harvey P. Dale, Sep 15 2021 *)

Crossrefs

Intersection of A005384 and A052291.

Sequence in context: A038526 A082755 A042067 * A338262 A376605 A042579

Adjacent sequences: A333800 A333801 A333802 * A333804 A333805 A333806

Keyword

nonn

Author

Robert Israel, Apr 05 2020

Status

approved