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Array (p(n,k)) read by antidiagonals: p(n,k) is the index of the prime in position (n,k) in the array A333086.
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%I #6 Mar 10 2020 23:49:36

%S 1,2,4,3,5,9,6,10,12,7,24,15,25,21,8,51,46,37,43,11,13,251,98,271,140,

%T 32,28,20,3121,329,1430,35505,231,40,93,22,42613,500,5185,85968,349,

%U 130,311,151,35

%N Array (p(n,k)) read by antidiagonals: p(n,k) is the index of the prime in position (n,k) in the array A333086.

%C As a sequence, this is a permutation of the positive integers.

%e Northwest corner:

%e 1 2 3 6 24 51

%e 4 5 10 15 46 98

%e 9 12 25 37 271 1430

%e 7 21 43 140 35505 85968

%e 8 11 32 231 349 4410

%e 13 28 40 130 5655 20908

%e The 4th prime is 7, which occurs in the position (2,1) in A333086, so that p(2,1) = 4.

%t W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];

%t t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 100}];

%t u = Flatten[Position[t, 1]] ; v[n_, k_] := W[u[[n]], k];

%t p[n_] := Table[v[n, k], {k, 1, 40}];

%t TableForm[Table[Select[p[n], PrimeQ], {n, 1, 10}]]

%t t1 = Table[PrimePi[Select[p[n], PrimeQ]], {n, 1, 10}]

%t tt[n_, k_] := t1[[n]][[k]];

%t Table[tt[n, k], {n, 1, 10}, {k, 1, 10}] (* A333087 array *)

%t ttt = Table[tt[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* A333087 sequence *)

%Y Cf. A000040, A099000 (row 1), A333028, A333086.

%K nonn,tabl,hard

%O 1,2

%A _Clark Kimberling_, Mar 10 2020