A344448 - OEIS

%I #47 Aug 08 2022 20:31:18

%S 2,3,2,5,3,2,17,11,11,191,41,191,263,311,4457,59,269,557,557,5867,

%T 3803,71,383,761,659,7001,13859,1889,89,509,797,887,7019,22961,16829,

%U 17,101,809,863,1607,7541,31223,62549,69677,113921,131,827,977,2309,8609,44351,67103,102647,176459,24071

%N 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.

%C 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.

%C 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.

%H J. Brillhart, D. H. Lehmer and J. L. Selfridge, <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0384673-1">New primality criteria and factorizations of 2^m+-1</a>, Math. Compl. 29 (1975) 620-647.

%e Array begins:

%e ===============================================================

%e n\k |   0    1    2    3     4     5      6      7      8     9

%e ----+----------------------------------------------------------

%e   1 |   2    2    2  191  4457  3803   1889     17 113921 24071

%e   2 |   3    3   11  311  5867 13859  16829  69677 176459 ...

%e   3 |   5   11  263  557  7001 22961  62549 102647 ...

%e   4 |  17  191  557  659  7019 31223  67103 164963 ...

%e   5 |  41  269  761  887  7541 44351 181931 170669 ...

%e   6 |  59  383  797 1607  8609 45737 188333 207923 ...

%e   7 |  71  509  863 2309  8627 61751 205433 235679 ...

%e   8 |  89  809  977 2621 21773 63377 210407 342833 ...

%e   9 | 101  827 1091 2687 22871 79481 219761 459209 ...

%o (PARI) 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

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

%Y Cf. A342690, A342691.

%K nonn,tabl

%O 1,1

%A _Martin Becker_, May 19 2021