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

1,2

Links

Table of n, a(n) for n=1..47.

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.

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        5         12          23                44
------------------------------------------------------------------------

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}]

Crossrefs

Cf. A010051, A344316.

Sequence in context: A000327 A220425 A130624 * A066869 A023172 A270681

Adjacent sequences: A344843 A344844 A344845 * A344847 A344848 A344849

Keyword

nonn

Author

Wesley Ivan Hurt, May 29 2021

Status

approved