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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.
0
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.
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
KEYWORD
nonn
AUTHOR
Saish S. Kambali, Nov 12 2022
EXTENSIONS
More terms from Gerry Martens, Nov 12 2022
STATUS
approved