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

1,4

Links

Table of n, a(n) for n=1..62.

Formula

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

Example

                                                      [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}
------------------------------------------------------------------------

Mathematica

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

Crossrefs

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

Sequence in context: A213064 A076466 A073117 * A039879 A125204 A241386

Adjacent sequences: A342692 A342693 A342694 * A342696 A342697 A342698

Keyword

nonn

Author

Wesley Ivan Hurt, May 18 2021

Status

approved