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A344448
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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.
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2
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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
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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 ...
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PROG
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(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
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CROSSREFS
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The first column T(n,0) is A053182(n). The second column T(n,1) is A066100(n).
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KEYWORD
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AUTHOR
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STATUS
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approved
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