A359298 Array T(n, k) read by antidiagonals: for n >= 0 and k >= 0, row n lists the positive integers m such that m - k is prime or 1, and m - h, for 0 <= h < k, is not prime.

1, 2, 4, 3, 6, 9, 5, 8, 15, 10, 7, 12, 21, 16, 27, 11, 14, 25, 22, 35, 28, 13, 18, 33, 26, 51, 36, 95, 17, 20, 39, 34, 57, 52, 119, 96, 19, 24, 45, 40, 65, 58, 145, 120, 121, 23, 30, 49, 46, 77, 66, 187, 146, 147, 122, 29, 32, 55, 50, 87, 78, 205, 188, 189

Offset: 1

Author

Clark Kimberling, Jan 01 2023

Keywords

• nonn
• tabl

Comments

Essentially, for n >= 0, row n lists the numbers whose distance down to the nearest prime is n.

Examples

			Corner:
   1     2     3     5     7    11    13    17     19     23     29
   4     6     8    12    14    18    20    24     30     32     38
   9    15    21    25    33    39    45    49     55     63     69
  10    16    22    26    34    40    46    50     56     64     70
  27    35    51    57    65    77    87    93    117    135    143
  28    36    52    58    66    78    88    94    118    136    144
Row 0 includes 19 because 19 is prime, and 19 - 19 = 0.
Row 1 includes 8 because the nearest prime down from 8 is 7, and 8 - 7 = 1.
		

Crossrefs

Cf. A000040, A008578, A359299, A359300.

Programs

Mathematica

rows = 15;
row[0] = Join[{1}, Map[Prime, Range[250]]];
Table[row[z] = Map[#[[1]] &, Select[Map[{#, Apply[And,
Join[{MemberQ[row[0], # - z]}, Table[! MemberQ[row[0], # - k], {k, 0, z - 1}]]]} &, Range[Max[row[z - 1]]]], #[[2]] &]], {z, rows}];
Table[row[z], {z, 0, rows}] // ColumnForm  (* A359298 array *)
t[n_, k_] := row[n - 1][[k]];
u = Table[t[n - k + 1, k], {n, 15}, {k, n, 1, -1}] //
Flatten  (* A359298 sequence *)
(* Peter J. C. Moses Dec 18 2022 *)