A342691 Primes of the form (p^k)^2 + p^k + 1 with prime p and positive integer k.

7, 13, 31, 73, 307, 757, 1723, 3541, 5113, 8011, 10303, 17293, 28057, 30103, 86143, 147073, 262657, 459007, 492103, 552793, 579883, 598303, 684757, 704761, 735307, 830833, 1191373, 1204507, 1353733, 1395943, 1424443, 1482307, 1772893, 1886503, 2037757, 2212657

Offset: 1

Author

Martin Becker, May 18 2021

Keywords

• nonn

Comments

Also, primes of the form (p^3^m)^2 + p^3^m + 1 with prime p and nonnegative integer m, since k must be a power of 3, from the theory of cyclotomic polynomials.

Examples

			31 = (5^1)^2 + 5^1 + 1 is in the sequence as 31 is prime and 5 is prime and 1 is a positive integer.
73 = (2^3)^2 + 2^3 + 1 is in the sequence as it is prime and 2 is prime and 3 is a positive integer.
		

Links

• Martin Becker, Table of n, a(n) for n = 1..20000

Crossrefs

Contains A053183 and A063784.

Intersection of A335865 and A000040 minus {3}.

Cf. A342690, A344448.

Programs

Mathematica

Select[Table[q^2 + q + 1, {q, Select[Range[1500], PrimePowerQ[#] &]}], PrimeQ] (* Amiram Eldar, Aug 16 2024 *)

PARI

for(q=2,2048,if(isprimepower(q),m=q^2+q+1;if(isprime(m),print1(m, ", "))))