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

1,1

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.

Links

Table of n, a(n) for n=1..51.

Example

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.

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 *)

Crossrefs

Cf. A105093.

Subsequence of A031918 (by definition).

Sequence in context: A041097 A067071 A031918 * A101231 A141949 A242675

Adjacent sequences: A358025 A358026 A358027 * A358029 A358030 A358031

Keyword

nonn

Author

Saish S. Kambali, Nov 12 2022

Extensions

More terms from Gerry Martens, Nov 12 2022

Status

approved