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

1,3

Links

Table of n, a(n) for n=1..43.

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.

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

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}]

Crossrefs

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

Sequence in context: A011845 A099450 A360191 * A145129 A001793 A325919

Adjacent sequences: A344844 A344845 A344846 * A344848 A344849 A344850

Keyword

nonn

Author

Wesley Ivan Hurt, May 29 2021

Status

approved