A387386 Array read by antidiagonals, where the n-th row lists primes p such that p, p+x_1, ..., p+x_1+...+x_k are consecutive primes, where (x_1, ..., x_k) is the n-th composition corresponding to an admissible prime tuple pattern (see A387383).

2, 3, 3, 5, 5, 7, 7, 11, 13, 23, 11, 17, 19, 31, 7, 13, 29, 37, 47, 13, 5, 17, 41, 43, 53, 37, 11, 89, 19, 59, 67, 61, 67, 17, 359, 23, 23, 71, 79, 73, 97, 41, 389, 53, 29, 29, 101, 97, 83, 103, 101, 401, 131, 59, 5, 31, 107, 103, 131, 193, 107, 449, 173, 71, 11, 139

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) | A029710

4 | (0,6) | A031924

5 | (0,4,6) | A022005

6 | (0,2,6) | A022004

7 | (0,8) | A031926

8 | (0,6,8) | A049438

9 | (0,2,8) | A049437 (except first term 3)

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

11 | (0,10) | A031928

12 | (0,6,10) | A078562

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 |   7  13  19  37  43   67   79   97  103  109  127  163
   4 |  23  31  47  53  61   73   83  131  151  157  167  173
   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 |  89 359 389 401 449  479  491  683  701  719  743  761
   8 |  23  53 131 173 233  263  563  593  653 1013 1223 1283
   9 |  29  59  71 149 269  431  569  599 1031 1061 1229 1289
  10 |   5  11 101 191 821 1481 1871 2081 3251 3461 5651 9431
  11 | 139 181 241 283 337  409  421  547  577  631  691  709
  12 |  31  61  73 157 271  373  433  607  733  751 1291 1543

Crossrefs

Cf. A387383, A387384 (not necessarily consecutive primes), A387387 (first column).

Sequence in context: A257539 A159050 A363433 * A210336 A280271 A145834

Adjacent sequences: A387383 A387384 A387385 * A387387 A387388 A387389

Keyword

nonn,tabl

Author

Pontus von Brömssen, Aug 29 2025

Status

approved