A333086 -id:A333086 - cp's OEIS frontend

A333087 Array (p(n,k)) read by antidiagonals: p(n,k) is the index of the prime in position (n,k) in the array A333086.

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, 32, 28, 20, 3121, 329, 1430, 35505, 231, 40, 93, 22, 42613, 500, 5185, 85968, 349, 130, 311, 151, 35

Offset: 1

Clark Kimberling, Mar 10 2020

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

			Northwest corner:
   1   2   3    6    24     51
   4   5  10   15    46     98
   9  12  25   37   271   1430
   7  21  43  140 35505  85968
   8  11  32  231   349   4410
  13  28  40  130  5655  20908
The 4th prime is 7, which occurs in the position (2,1) in A333086, so that p(2,1) = 4.

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

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 100}];
u = Flatten[Position[t, 1]] ; v[n_, k_] := W[u[[n]], k];
p[n_] := Table[v[n, k], {k, 1, 40}];
TableForm[Table[Select[p[n], PrimeQ], {n, 1, 10}]]
t1 = Table[PrimePi[Select[p[n], PrimeQ]], {n, 1, 10}]
tt[n_, k_] := t1[[n]][[k]];
Table[tt[n, k], {n, 1, 10}, {k, 1, 10}]  (* A333087 array *)
ttt = Table[tt[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten  (* A333087 sequence *)

A333028 Array consisting of the primitive rows of the Wythoff array (A035513), read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 14, 5, 11, 23, 17, 8, 18, 37, 28, 19, 13, 29, 60, 45, 31, 25, 21, 47, 97, 73, 50, 41, 27, 34, 76, 157, 118, 81, 66, 44, 30, 55, 123, 254, 191, 131, 107, 71, 49, 35, 89, 199, 411, 309, 212, 173, 115, 79, 57, 43, 144, 322, 665, 500, 343, 280

Offset: 1

Clark Kimberling, Mar 10 2020

In a row of the Wythoff array, either every two consecutive terms are relatively prime or else no two consecutive terms are relatively prime. In the first case, we call the row primitive; otherwise, the row is an integer multiple of a tail of a preceding row. The primitive rows are interspersed, in the sense that if h < k then the numbers in row k are interspersed, in magnitude, among numbers in row h. In each row, every pair of consecutive numbers is a Wythoff pair of relatively prime numbers. The array includes every prime.

			Northwest corner:
   1   2   3    5    8   13  21    34
   4   7  11   18   29   47  76   123
  14  23  37   60   97  157  254  411
  17  28  45   73  118  191  309  500
  19  31  50   81  131  212  343  555
  25  41  66  107  173  280  453  733
  27  44  71  115  186  301  487  788
  30  49  79  128  207  335  542  877

Cf. A000045, A000032, A332937, A332938, A333029, A333086.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 160}]
u = Flatten[Position[t, 1]]; v[n_, k_] := W[u[[n]], k];
TableForm[Table[v[n, k], {n, 1, 30}, {k, 1, 8}]] (* A333028 array *)
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* A333028 sequence *)

cp's OEIS Frontend

A333087 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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