A344448 -id:A344448 - cp's OEIS frontend

A342690 Prime powers q in A246655 such that q^2 + q + 1 is prime.

2, 3, 5, 8, 17, 27, 41, 59, 71, 89, 101, 131, 167, 173, 293, 383, 512, 677, 701, 743, 761, 773, 827, 839, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1331, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663

Offset: 1

Martin Becker, May 18 2021

nonn

Also, prime powers q = p^(3^k) with prime p and nonnegative integer k and the property that q^2 + q + 1 is prime, since the exponent must be a power of 3, from the theory of cyclotomic polynomials. 17^(3^7) is in the sequence, generating a 5382-digit prime.

			5 = 5^1 is a term: 5^2 + 5 + 1 = 31 is prime.
8 = 2^3 is a term: 8^2 + 8 + 1 = 73 is prime.

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

Intersection of A246655 and A002384.

Cf. A342691, A344448.

Select[Range@2000,PrimePowerQ@#&&PrimeQ[#^2+#+1]&] (* Giorgos Kalogeropoulos, May 18 2021 *)

N=50; i=0; a=vector(N); for(q=2, oo, if(isprimepower(q) && isprime(q^2+q+1), i+=1; a[i]=q; if(i==N, break))); a

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

Original entry on oeis.org

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

Martin Becker, May 18 2021

nonn

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.

			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.

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

Contains A053183 and A063784.
Intersection of A335865 and A000040 minus {3}.
Cf. A342690, A344448.

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

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

cp's OEIS Frontend

A342690 Prime powers q in A246655 such that q^2 + q + 1 is prime.

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