A344846 Sum of the prime numbers appearing along the border of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows.

0, 5, 12, 23, 44, 80, 136, 195, 225, 329, 320, 694, 791, 808, 899, 953, 1378, 2485, 1905, 2152, 2898, 3364, 2577, 4913, 4061, 5589, 4638, 6978, 5432, 10814, 5305, 10157, 9135, 10507, 10976, 15342, 5149, 14352, 16891, 17827, 11327, 26086, 14738, 19337, 23838, 30784, 16701

Offset: 1

Author

Wesley Ivan Hurt, May 29 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        5         12          23                44
------------------------------------------------------------------------
		

Crossrefs

Cf. A010051, A344316.

Programs

Mathematica

Table[Sum[(n^2 - k + 1) (PrimePi[n^2 - k + 1] - PrimePi[n^2 - k]) + k (PrimePi[k] - PrimePi[k - 1]), {k, n}] + Sum[(n*j + 1) (PrimePi[n*j + 1] - PrimePi[n*j]), {j, n - 2}], {n, 60}]

Formula

a(n) = Sum_{k=1..n} ((n^2-k+1) * c(n^2-k+1) + k * c(k)) + Sum_{k=1..n-2} ((n*k+1) * c(n*k+1)), where c(n) is the prime characteristic.