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Rectangular array by antidiagonals: R(m,n) = least k such that 2n*prime(m)^k - 1 is prime.
3

%I #4 Nov 04 2024 22:29:37

%S 1,1,1,1,1,4,2,1,1,1,1,1,1,1,2,1,1,2,4,1,2,4,2,3,1,2,2,2,1,1,1,1,1,1,

%T 1,1,2,1,2,2,1,2,1,1,6,2,1,1,1,2,1,2,1,5,1,1,1,1,1,1,1,2,1,1,3,1,1,2,

%U 2,1,4,10,4,6,2,1,2,1

%N Rectangular array by antidiagonals: R(m,n) = least k such that 2n*prime(m)^k - 1 is prime.

%e Corner:

%e 1 1 1 2 1 1 4 1 2 2 1 1 2 3 1 2

%e 1 1 1 1 1 2 1 1 1 1 2 1 2 1 1 3

%e 4 1 1 2 3 1 2 1 1 2 1 2 4 1 1 8

%e 1 1 4 1 1 2 1 1 1 3 1 1 2 1 1 2

%e 2 1 2 1 1 2 1 4 1 1 1 3 18 1 2 1

%e 2 2 1 2 1 1 10 1 1 1 2 1 2 6 1 2

%e 2 1 1 2 2 4 1 4 2 1 3 7 1 4 1 8

%e 1 1 1 1 6 6 1 2 3 1 1 1 11 1 6 1

%e 6 5 1 2 1 1 2 4 7 1 2 1 2 1 3 4

%e 1 3 1 10 1 1 2 1 2 6 2 1 2 5 8 2

%e 1 2 1 1 1 1 1 1 9 1 2 1 1 2 3 2

%e 1 3 4 1 3 14 1 1 2 1 8 7 2 1 1 3

%e 2 1 1 1 4 2 3 1 1 2 14 7 1 6 2 1

%e 2 1 1 1 1 1 1 1 2 7 3 1 4 3 1 3

%e 4 1 1 1 2 2 1 3 1 2 1 7 8 1 1 1

%e 1 1 8 15 1 2 4 1 9 4 1 1 2 1 1 2

%t f[m_, n_, k_] := 2 n*Prime[m]^k - 1;

%t s[m_, n_] := Select[Range[20], PrimeQ[f[m, n, #]] &, 1]

%t u[m_] := Flatten[Table[s[m, n], {n, 1, 60}]]

%t Column[Table[Take[u[m], 16], {m, 1, 16}]]

%t r[m_] := Take[u[m], 12]; w[m_, n_] := r[m][[n]];

%t Table[w[m, n], {m, 1, 16}, {n, 1, 12}] (* array *)

%t Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)

%Y Cf. A000040, A377367.

%K nonn,tabl

%O 1,6

%A _Clark Kimberling_, Oct 31 2024