A387384 Array read by antidiagonals, where the n-th row lists primes p such that p+x_1, ..., p+x_1+...+x_k are prime, where (x_1, ..., x_k) is the n-th composition corresponding to an admissible prime tuple pattern (see A387383).

2, 3, 3, 5, 5, 3, 7, 11, 7, 5, 11, 17, 13, 7, 7, 13, 29, 19, 11, 13, 5, 17, 41, 37, 13, 37, 11, 3, 19, 59, 43, 17, 67, 17, 5, 5, 23, 71, 67, 23, 97, 41, 11, 11, 3, 29, 101, 79, 31, 103, 101, 23, 23, 5, 5, 31, 107, 97, 37, 193, 107, 29, 53, 11, 11, 3

Offset

1,1

Comments

The first Hardy-Littlewood conjecture (or the k-tuple conjecture) implies that each row has an infinite number of terms. To ensure that the array is well-defined even if the conjecture is false, fill the rest of a row with 0's if there are only finitely many prime tuples for the corresponding pattern.

Rows include:

n | pattern | sequence

----+-----------+---------

1 | (0) | A000040

2 | (0,2) | A001359

3 | (0,4) | A023200

4 | (0,6) | A023201

5 | (0,4,6) | A022005

6 | (0,2,6) | A022004

7 | (0,8) | A023202

8 | (0,6,8) | A046138

9 | (0,2,8) | A046134

10 | (0,2,6,8) | A007530

11 | (0,10) | A023203

12 | (0,6,10) | A046139

Of course, all other admissible prime tuple patterns also appear as rows in this array.

Links

Pontus von Brömssen, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals)

Wikipedia, First Hardy-Littlewood conjecture.

Wikipedia, Prime k-tuple.

Example

Array begins:
  n\k| 1  2   3   4   5    6    7    8    9   10   11   12
  ---+----------------------------------------------------
   1 | 2  3   5   7  11   13   17   19   23   29   31   37
   2 | 3  5  11  17  29   41   59   71  101  107  137  149
   3 | 3  7  13  19  37   43   67   79   97  103  109  127
   4 | 5  7  11  13  17   23   31   37   41   47   53   61
   5 | 7 13  37  67  97  103  193  223  277  307  457  613
   6 | 5 11  17  41 101  107  191  227  311  347  461  641
   7 | 3  5  11  23  29   53   59   71   89  101  131  149
   8 | 5 11  23  53 101  131  173  191  233  263  563  593
   9 | 3  5  11  29  59   71  101  149  191  269  431  569
  10 | 5 11 101 191 821 1481 1871 2081 3251 3461 5651 9431
  11 | 3  7  13  19  31   37   43   61   73   79   97  103
  12 | 7 13  31  37  61   73   97  103  157  223  271  307

Crossrefs

Cf. A292224, A387383, A387385 (first column), A387386 (consecutive primes).

Sequence in context: A046146 A393607 A081768 * A273493 A193404 A072923

Adjacent sequences: A387381 A387382 A387383 * A387385 A387386 A387387

Keyword

nonn,tabl

Author

Pontus von Brömssen, Aug 29 2025

Status

approved