A359299 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, 10, 15, 8, 7, 12, 21, 14, 25, 11, 16, 27, 20, 33, 24, 13, 18, 35, 26, 49, 32, 91, 17, 22, 39, 34, 55, 48, 121, 90, 19, 28, 45, 38, 63, 54, 143, 120, 119, 23, 30, 51, 44, 75, 62, 185, 142, 141, 118, 29, 36, 57, 50, 85, 74, 205, 184, 183

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     10    12    16    18    22    28    30      36     40
    9    15     21    27    35    39    45    51    57      65     69
    8    14     20    26    34    38    44    50    56      64     68
   25    33     49    55    63    75    85    93    123    133    145
   24    32     48    54    62    74    84    92    122    132    144
Row 0 includes 19 because 19 is prime, and 19 - 19 = 0.
Row 1 includes 10 because the nearest prime up from 10 is 11, and 11 - 10 = 1.
		

Crossrefs

Cf. A000040, A008578, A359298, 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   (* A359299 array *)
t[n_, k_] := row[n - 1][[k]]
u = Table[t[n - k + 1, k], {n, 15}, {k, n, 1, -1}] //
Flatten  (* A359299 sequence *)
(* Peter J. C. Moses Dec 18 2022 *)