%I #6 May 29 2021 20:32:41
%S 0,5,12,23,44,80,136,195,225,329,320,694,791,808,899,953,1378,2485,
%T 1905,2152,2898,3364,2577,4913,4061,5589,4638,6978,5432,10814,5305,
%U 10157,9135,10507,10976,15342,5149,14352,16891,17827,11327,26086,14738,19337,23838,30784,16701
%N 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.
%F 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.
%e [1 2 3 4 5]
%e [1 2 3 4] [6 7 8 9 10]
%e [1 2 3] [5 6 7 8] [11 12 13 14 15]
%e [1 2] [4 5 6] [9 10 11 12] [16 17 18 19 20]
%e [1] [3 4] [7 8 9] [13 14 15 16] [21 22 23 24 25]
%e ------------------------------------------------------------------------
%e n 1 2 3 4 5
%e ------------------------------------------------------------------------
%e a(n) 0 5 12 23 44
%e ------------------------------------------------------------------------
%t 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}]
%Y Cf. A010051, A344316.
%K nonn
%O 1,2
%A _Wesley Ivan Hurt_, May 29 2021