A342695 a(n) is the number of primes in an n X n square array that do not appear on its border, with the elements of the square array being the numbers from 1..n^2, listed in increasing order by rows.

0, 0, 1, 2, 4, 4, 8, 10, 14, 15, 21, 21, 27, 31, 36, 42, 48, 46, 58, 61, 68, 73, 83, 83, 96, 100, 110, 114, 127, 123, 144, 146, 157, 165, 175, 179, 201, 201, 212, 221, 241, 235, 258, 265, 275, 282, 303, 301, 328, 330, 346, 351, 381, 377, 403, 406, 427, 433, 455, 452, 486, 493

Offset: 1

Author

Wesley Ivan Hurt, May 18 2021

Keywords

• nonn

Examples

			                                                      [1   2  3  4  5]
                                      [1   2  3  4]   [6   7  8  9 10]
                            [1 2 3]   [5   6  7  8]   [11 12 13 14 15]
                   [1 2]    [4 5 6]   [9  10 11 12]   [16 17 18 19 20]
           [1]     [3 4]    [7 8 9]   [13 14 15 16]   [21 22 23 24 25]
------------------------------------------------------------------------
  n         1        2         3            4                 5
------------------------------------------------------------------------
  a(n)      0        0         1            2                 4
------------------------------------------------------------------------
  primes   {}       {}        {5}        {7,11}         {7,13,17,19}
------------------------------------------------------------------------
		

Crossrefs

Cf. A000720 (pi), A038107, A221490, A344316 (on border), A344349.

Programs

Mathematica

Table[PrimePi[n*(n - 1)] - PrimePi[n] - Sum[PrimePi[n*k + 1] - PrimePi[n*k], {k, n - 2}], {n, 100}]

Formula

a(n) = pi(n*(n-1)) - pi(n) - Sum_{k=1..n-2} (pi(n*k+1) - pi(n*k)).