A344847 Sum of the prime numbers in, but not on 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, 0, 5, 18, 56, 80, 192, 306, 566, 731, 1273, 1433, 2123, 3023, 3762, 5128, 6604, 7038, 9694, 11735, 13942, 16695, 21015, 22027, 28292, 31972, 37830, 41516, 50405, 51983, 64936, 70032, 80537, 90331, 100611, 108869, 130965, 134475, 149660, 165879, 191969, 196185, 223782

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        0         5           18                56
------------------------------------------------------------------------
		

Crossrefs

Cf. A010051, A344316, A344846 (sum of primes on border).

Programs

Mathematica

Table[Sum[i (PrimePi[i] - PrimePi[i - 1]), {i, n^2}] - 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^2} k * c(k)) - (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.