A337767 Array T(n,k) (n >= 1, k >= 1) read by upward antidiagonals and defined as follows. Let N(p,i) denote the result of applying "nextprime" i times to p; T(n,k) = smallest prime p such that N(p,n) - p = 2*k, or 0 if no such prime exists.

3, 0, 7, 0, 3, 23, 0, 0, 5, 89, 0, 0, 0, 23, 139, 0, 0, 0, 3, 19, 199, 0, 0, 0, 0, 7, 47, 113, 0, 0, 0, 0, 3, 17, 83, 1831, 0, 0, 0, 0, 0, 5, 23, 211, 523, 0, 0, 0, 0, 0, 0, 17, 43, 109, 887, 0, 0, 0, 0, 0, 0, 3, 13, 79, 317, 1129, 0, 0, 0, 0, 0, 0, 0, 7, 19, 107, 619, 1669

Offset

1,1

Comments

The positive entries in each row and column are distinct.

Number of zeros right of 3 are 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 3, 3, 3, 6, 5, 5, 4, 6, ..., .

Number of zeros in the n-th row are 0, 1, 3, 4, 6, 7, 10, 13, 14, 17, 18, 20, 22, 25, 28, 30, 32, 36, 37, 40, 45, 47, 51, 52, 55, ..., .

The usual convention in the OEIS is to use -1 in the "escape clause" - that is, when "no such terms exists". It is probably too late to change this sequence, but it should not be cited as a role model for other sequences. - N. J. A. Sloane, Jan 19 2021

a(1416), a(1637), and a(1753) were provided by Brian Kehrig. - Martin Raab, Jun 28 2024

Links

Martin Raab, Table of n, a(n) for n = 1..1830 (antidiagonals 1..60)

Formula

T(n,k) = 0 if prime(n+2)-5 <= 2k. A089038.

T(n,k) = 3 if prime(n+2) = 2k+6. A067076.

Example

The initial rows of the array are:
  3, 7, 23, 89, 139, 199, 113, 1831, 523, 887, 1129, 1669, 2477, 2971, 4297, ...
  0, 3, 5, 23, 19, 47, 83, 211, 109, 317, 619,  199, 1373, 1123, 1627, 4751, ...
  0, 0, 0,  3,  7, 17, 23,  43,  79, 107, 109,  113,  197,  199,  317,  509, ...
  0, 0, 0,  0,  3,  5, 17,  13,  19,  47,  79,   73,  113,  109,  193,  317, ...
  0, 0, 0,  0,  0,  0,  3,   7,  11,  17,  19,   43,   71,   73,  107,  191, ...
  0, 0, 0,  0,  0,  0,  0,   3,   5,  11,   7,   13,   41,   31,   67,  107, ...
  0, 0, 0,  0,  0,  0,  0,   0,   0,   3,   0,    5,   11,   13,   23,   47, ...
  0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,    0,    3,    0,    7,   29, ...
  0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,    0,    0,    3,    0,    5, ...
The initial antidiagonals are:
  [3]
  [0, 7]
  [0, 3, 23]
  [0, 0, 5, 89]
  [0, 0, 0, 23, 139]
  [0, 0, 0, 3, 19, 199]
  [0, 0, 0, 0, 7, 47, 113]
  [0, 0, 0, 0, 3, 17, 83, 1831]
  [0, 0, 0, 0, 0, 5, 23, 211, 523]
  [0, 0, 0, 0, 0, 0, 17, 43, 109, 887]
  [0, 0, 0, 0, 0, 0, 3, 13, 79, 317, 1129]
  ...

Mathematica

t[r_, c_] := If[ 2c <= Prime[r + 2] - 5, 0, Block[{p = 3}, While[ NextPrime[p, r] != 2c + p && p < 52000000, p = NextPrime@ p]; If[p > 52000000, 0, p]]]; Table[ t[r -c +1, c], {r, 11}, {c, r}] // Flatten

Crossrefs

Cf. A000230, A144103, A339943, A339944 (rows 1 to 4), A086153.

Sequence in context: A099893 A135534 A346516 * A249904 A324875 A266437

Adjacent sequences: A337764 A337765 A337766 * A337768 A337769 A337770

Keyword

nonn,tabl

Author

Robert G. Wilson v, Sep 19 2020

Extensions

Entry revised by N. J. A. Sloane, Nov 07 2020

Deleted a-file and b-file because entries were unreliable. - N. J. A. Sloane, Nov 01 2021

Status

approved