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

1,1

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.

Links

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Contains A053183 and A063784.

Intersection of A335865 and A000040 minus {3}.

Cf. A342690, A344448.

Sequence in context: A174217 A040061 A289123 * A215932 A091432 A102002

Adjacent sequences: A342688 A342689 A342690 * A342692 A342693 A342694

Keyword

nonn

Author

Martin Becker, May 18 2021

Status

approved