A358028 Primes p = prime(9*t+1) such that the 9 consecutive primes prime(9*t+1) .. prime(9*t+9) arranged in a 3 X 3 array have at least 2 equal sums along the rows, columns or main diagonals.

2, 29, 67, 107, 157, 257, 311, 367, 541, 599, 709, 769, 829, 967, 1021, 1549, 1741, 1811, 1879, 1973, 2609, 2677, 3019, 3541, 3677, 4051, 4217, 4271, 4517, 4597, 4663, 4931, 5227, 5303, 5399, 5449, 5623, 5683, 5839, 6079, 6229, 6301, 6361, 6451, 6949, 7253, 7351, 7477, 7537, 7589, 7673

Offset: 1

Author

Saish S. Kambali, Nov 12 2022

Keywords

• nonn

Comments

Primes are taken in successive blocks of 9 and arranged, for t>=0,

| prime(9*t+1) | prime(9*t+2) | prime(9*t+3) |

| prime(9*t+4) | prime(9*t+5) | prime(9*t+6) |

| prime(9*t+7) | prime(9*t+8) | prime(9*t+9) |

There are 8 lines altogether: 3 rows, 3 columns, and 2 main diagonals.

The sum of the first row is never duplicated since any other line has a greater sum.

The sum of the last row is never duplicated since any other line has a smaller sum.

Examples

			2 is a term since its block of 9 primes is
  | 2  | 3  | 5  |
  | 7  | 11 | 13 |
  | 17 | 19 | 23 |
which has among its lines (3 + 11 + 19) = (17 + 11 + 5).
67 is a term since its block of 9 primes (the 3rd block) is 67..103,
  | 67 | 71 | 73 |
  | 79 | 83 | 89 |
  | 97 | 101| 103|
which has 67+83+103 = 97+83+73.
		

Crossrefs

Cf. A105093.

Subsequence of A031918 (by definition).

Programs

Mathematica

a = {}
row = {{1, 4, 7}, {2, 5, 8}, {3, 6, 9}};
col = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};
dia = {{1, 3}, {5, 5}, {9, 7}};
Duplicates[l_] :=
Block[{i}, i[n_] := (i[n] = n; Unevaluated@Sequence[]); i /@ l]
Do[If[Duplicates[
    Flatten[{Total[Prime[row + 9 n]], Total[Prime[col + 9 n]],
      Total[Prime[dia + 9 n]]}]] != {},
  AppendTo[a, Prime[9 n + 1]]], {n, 0, 110}]
a (* Gerry Martens, Nov 12 2022 *)

Extensions

More terms from Gerry Martens, Nov 12 2022