A342690 -id:A342690 - cp's OEIS frontend

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

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, ", "))))

A344448 Square array read by antidiagonals upwards: T(n,k) for integer k >= 0 is the n-th prime p such that p^(2*3^k) + p^(3^k) + 1 is prime.

Original entry on oeis.org

2, 3, 2, 5, 3, 2, 17, 11, 11, 191, 41, 191, 263, 311, 4457, 59, 269, 557, 557, 5867, 3803, 71, 383, 761, 659, 7001, 13859, 1889, 89, 509, 797, 887, 7019, 22961, 16829, 17, 101, 809, 863, 1607, 7541, 31223, 62549, 69677, 113921, 131, 827, 977, 2309, 8609, 44351, 67103, 102647, 176459, 24071

Offset: 1

Martin Becker, May 19 2021

T(n,k)^(3^k), for all n >= 1, k >= 0, arranged by increasing values, is A342690. It is conjectured that all columns are infinite. If 3^k was replaced by k in the definition, all additional columns would be empty, as x^(2*k) + x^k + 1 is reducible if k has prime factors other than 3. For checking the property, Pocklington-Lehmer type primality tests seem particularly effective, as n-1 always has a large smooth factor p^(3^k), cf. the paper of Brillhart, Lehmer and Selfridge (1975), Theorem 5.

This array describes the essence of A342690 and A342691 in much more terse form. T(1, 8) = 113921 matches the 33177-digit value q = 113921^3^8 in A342690 and the 66353-digit prime q^2+q+1 in A342691.

			Array begins:
===============================================================
n\k |   0    1    2    3     4     5      6      7      8     9
----+----------------------------------------------------------
  1 |   2    2    2  191  4457  3803   1889     17 113921 24071
  2 |   3    3   11  311  5867 13859  16829  69677 176459 ...
  3 |   5   11  263  557  7001 22961  62549 102647 ...
  4 |  17  191  557  659  7019 31223  67103 164963 ...
  5 |  41  269  761  887  7541 44351 181931 170669 ...
  6 |  59  383  797 1607  8609 45737 188333 207923 ...
  7 |  71  509  863 2309  8627 61751 205433 235679 ...
  8 |  89  809  977 2621 21773 63377 210407 342833 ...
  9 | 101  827 1091 2687 22871 79481 219761 459209 ...

J. Brillhart, D. H. Lehmer and J. L. Selfridge, New primality criteria and factorizations of 2^m+-1, Math. Compl. 29 (1975) 620-647.

The first column T(n,0) is A053182(n). The second column T(n,1) is A066100(n).

Cf. A342690, A342691.

N=5; K=2; m=matrix(N, K+1); for(k=0, K, i=0; forprime(p=2, , q=p^3^k;if(isprime(q^2+q+1, 1), i+=1; m[i,k+1]=p; if(i==N, break)))); m

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI