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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.

%I A344847 #5 May 29 2021 20:31:16
%S A344847 0,0,5,18,56,80,192,306,566,731,1273,1433,2123,3023,3762,5128,6604,
%T A344847 7038,9694,11735,13942,16695,21015,22027,28292,31972,37830,41516,
%U A344847 50405,51983,64936,70032,80537,90331,100611,108869,130965,134475,149660,165879,191969,196185,223782
%N 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.
%F A344847 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.
%e A344847                                                       [1   2  3  4  5]
%e A344847                                       [1   2  3  4]   [6   7  8  9 10]
%e A344847                             [1 2 3]   [5   6  7  8]   [11 12 13 14 15]
%e A344847                    [1 2]    [4 5 6]   [9  10 11 12]   [16 17 18 19 20]
%e A344847            [1]     [3 4]    [7 8 9]   [13 14 15 16]   [21 22 23 24 25]
%e A344847 ------------------------------------------------------------------------
%e A344847   n         1        2         3            4                 5
%e A344847 ------------------------------------------------------------------------
%e A344847   a(n)      0        0         5           18                56
%e A344847 ------------------------------------------------------------------------
%t A344847 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}]
%Y A344847 Cf. A010051, A344316, A344846 (sum of primes on border).
%K A344847 nonn
%O A344847 1,3
%A A344847 _Wesley Ivan Hurt_, May 29 2021