A333028 -id:A333028 - cp's OEIS frontend

A333029 Order array of array A333028, read by antidiagonals.

1, 2, 4, 3, 6, 10, 5, 8, 15, 11, 7, 12, 24, 18, 13, 9, 19, 38, 28, 21, 16, 14, 30, 58, 45, 32, 25, 17, 22, 47, 77, 65, 50, 40, 27, 20, 35, 67, 95, 84, 70, 60, 44, 31, 23, 55, 86, 112, 102, 89, 79, 64, 49, 36, 26, 74, 104, 128, 118, 106, 97, 83, 69, 56, 43

Offset: 1

Clark Kimberling, Mar 10 2020

Suppose that (r(n,k)), for n >=1 and k >= 1 is an array of distinct numbers. Replace each r(n,k) by its position when all the numbers r(n,k) are ordered by <. The resulting array is the order array of (r(n,k)). The array A333029 is an interspersion and, as a sequence, a permutation of the natural numbers.

			Northwest corner:
   1   2   3   5   7    9   14   22
   4   6   8  12  19   30   47   67
  10  15  24  38  58   77   95  112
  11  18  24  45  65   84  102  118
  13  21  32  50  70   89  106  122
  16  25  40  60  79   97  114  130
  17  27  44  64  83  101  117  132

Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.

Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.

Cf. A333028.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 500}];
u = Flatten[Position[t, 1]] ; v[n_, k_] := W[u[[n]], k];
g = Sort[Table[v[n - k + 1, k], {n, 20}, {k, n, 1, -1}] // Flatten]
wo[n_, k_] := Length[Intersection[Range[v[n, k]], Complement[Range[1500], g]]]
WP[n_, k_] := v[n, k] - wo[n, k];
TableForm[Table[WP[n, k], {n, 1, 15}, {k, 1, 10}]]  (* A333029 array *)
Table[WP[n - k + 1, k], {n, 16}, {k, n, 1, -1}] // Flatten  (* A333029 sequence *)

A333086 Array read by antidiagonals: row n consists of the primes in row n of the array A333028.

Original entry on oeis.org

2, 3, 7, 5, 11, 23, 13, 29, 37, 17, 89, 47, 97, 73, 19, 233, 199, 157, 191, 31, 41, 1597, 521, 1741, 809, 131, 107, 71, 28657, 2207, 11933, 421493, 1453, 173, 487, 79, 514229, 3571, 50549, 1103483, 2351, 733, 2063, 877, 149, 433494437, 9349, 214129, 1785473

Offset: 1

Clark Kimberling, Mar 10 2020

The array shows, in order, the primes in the Wythoff array after deletion of all nonprimes. Every prime occurs exactly once; that is, every prime is uniquely expressible as F(k+1)*floor(n*tau) + (n-1)F(k), where tau = golden ratio (A001622), F = A000045 (Fibonacci numbers), and n and k are positive integers. We assume as true the conjecture that each row is infinite.

			Northwest corner:
    2    3    5    13      89      233
    7   11   29    47     199      521
   23   37   97   157    1741    11933
   17   73  191   809  421493  1103483
   19   31  131  1453    2351    42187
   41  107  173   733   55717   236021
Row 22 begins with 30631, 2187696161008162875319987.

Cf. A000040, A333028, A333087.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 200}];
u = Flatten[Position[t, 1]] ; v[n_, k_] := W[u[[n]], k];
p[n_] := Table[v[n, k], {k, 1, 1000}];
TableForm[Table[Select[p[n], PrimeQ], {n, 1, 100}]]

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.

Original entry on oeis.org

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 *)

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A333029 Order array of array A333028, read by antidiagonals.

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A333086 Array read by antidiagonals: row n consists of the primes in row n of the array A333028.

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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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