A344448 - OEIS

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.

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

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OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..55.

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

EXAMPLE

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 ...