A344316 Number of primes 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, 2, 3, 4, 5, 7, 7, 8, 8, 10, 9, 13, 12, 13, 12, 12, 13, 20, 14, 17, 17, 19, 16, 22, 18, 22, 19, 23, 19, 31, 18, 26, 24, 26, 25, 31, 18, 27, 28, 30, 22, 39, 25, 30, 31, 37, 26, 41, 29, 37, 32, 42, 28, 44, 31, 39, 30, 41, 32, 51, 33, 39, 40, 41, 36, 52, 35, 44, 39, 50, 39, 52, 39

Offset

1,2

Links

Table of n, a(n) for n=1..73.

Formula

a(n) = pi(n) + pi(n^2-1) - pi(n^2-n) + Sum_{k=1..n-2} (pi(n*k+1) - pi(n*k)).

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        2         3            4                 5
------------------------------------------------------------------------
  primes   {}      {2,3}    {2,3,7}    {2,3,5,13}       {2,3,5,11,23}
------------------------------------------------------------------------

Mathematica

Table[PrimePi[n] + PrimePi[n^2 - 1] - PrimePi[n*(n - 1)] + Sum[PrimePi[n*k + 1] - PrimePi[n*k], {k, n - 2}], {n, 100}]

Crossrefs

Cf. A000720 (pi), A038107, A221490, A344349.

Sequence in context: A336349 A255545 A034152 * A343271 A363897 A114707

Adjacent sequences: A344313 A344314 A344315 * A344317 A344318 A344319

Keyword

nonn

Author

Wesley Ivan Hurt, May 14 2021

Status

approved