A344316 -id:A344316 - cp's OEIS frontend

A221490 Number of primes of the form k*n + k - n, 1 <= k <= n.

0, 0, 1, 1, 3, 1, 2, 2, 5, 3, 6, 3, 5, 4, 4, 3, 9, 2, 6, 5, 8, 4, 9, 4, 9, 7, 10, 4, 17, 3, 10, 9, 11, 9, 15, 4, 9, 10, 13, 5, 20, 7, 11, 10, 16, 8, 19, 6, 18, 12, 17, 5, 23, 9, 18, 9, 15, 8, 26, 7, 15, 12, 16, 13, 29, 8, 18, 13, 26, 9, 25, 10, 19, 18, 16

Offset: 1

Reinhard Zumkeller, Jan 19 2013

nonn

Number of primes in n-th row of the triangle in A209297.

Number of primes along the main diagonal of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows (see square arrays in example). - Wesley Ivan Hurt, May 15 2021

			Row 10 of A209297 = [1,12,23,34,45,56,67,78,89,100] containing three primes: [23,67,89], therefore a(10) = 3;
row 11 of A209297 = [1,13,25,37,49,61,73,85,97,109,121] containing six primes: [13,37,61,73,97,109], therefore a(11) = 6.
From _Wesley Ivan Hurt_, May 15 2021: (Start)
                                                      [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         1            1                 3
------------------------------------------------------------------------
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A010051, A209297, A221491, A344316, A344349.

a221490 n = sum [a010051 (k*n + k - n) | k <- [1..n]]

Count[#,?PrimeQ]&/@Table[k*n+k-n,{n,75},{k,n}] (* _Harvey P. Dale, Apr 03 2015 *)

a(n) = sum(k=1, n, isprime(k*n + k - n)); \\ Michel Marcus, Jan 26 2022

a(n) = Sum_{k=1..n} A010051(A209297(n,k)).

a(n) = Sum_{k=1..n} c(n*(k-1)+k), where c is the prime characteristic. - Wesley Ivan Hurt, May 15 2021

A344349 Number of primes along the main antidiagonal of the n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows.

Original entry on oeis.org

0, 2, 3, 2, 3, 2, 6, 2, 3, 3, 6, 3, 7, 4, 7, 6, 6, 4, 10, 2, 8, 7, 9, 4, 11, 5, 10, 8, 11, 4, 17, 3, 10, 10, 12, 9, 16, 4, 10, 11, 14, 6, 21, 7, 11, 10, 16, 8, 19, 6, 19, 13, 17, 5, 25, 10, 19, 10, 16, 9, 27, 7, 16, 13, 16, 13, 31, 9, 18, 14, 27, 10, 26, 10, 20, 19, 17, 12, 30

Offset: 1

Wesley Ivan Hurt, May 15 2021

nonn

			                                                      [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            2                 3
------------------------------------------------------------------------
  primes   {}      {2,3}    {3,5,7}       {7,13}          {5,13,17}
------------------------------------------------------------------------

Cf. A010051, A221490, A344316 (primes along border).

Table[Sum[PrimePi[n*k - k + 1] - PrimePi[n*k - k], {k, n}], {n, 100}]

a(n) = Sum_{k=1..n} c(n*k-k+1), where c is the prime characteristic.

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.

Original entry on oeis.org

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

Wesley Ivan Hurt, May 29 2021

nonn

			                                                      [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
------------------------------------------------------------------------

Cf. A010051, A344316.

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}]

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.

A342695 a(n) is the number of primes in an n X n square array that do not appear on its border, with the elements of the square array being the numbers from 1..n^2, listed in increasing order by rows.

Original entry on oeis.org

0, 0, 1, 2, 4, 4, 8, 10, 14, 15, 21, 21, 27, 31, 36, 42, 48, 46, 58, 61, 68, 73, 83, 83, 96, 100, 110, 114, 127, 123, 144, 146, 157, 165, 175, 179, 201, 201, 212, 221, 241, 235, 258, 265, 275, 282, 303, 301, 328, 330, 346, 351, 381, 377, 403, 406, 427, 433, 455, 452, 486, 493

Offset: 1

Wesley Ivan Hurt, May 18 2021

nonn

			                                                      [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         1            2                 4
------------------------------------------------------------------------
  primes   {}       {}        {5}        {7,11}         {7,13,17,19}
------------------------------------------------------------------------

Cf. A000720 (pi), A038107, A221490, A344316 (on border), A344349.

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

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

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.

Original entry on oeis.org

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

Wesley Ivan Hurt, May 29 2021

nonn

			                                                      [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
------------------------------------------------------------------------

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

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}]

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.

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A221490 Number of primes of the form k*n + k - n, 1 <= k <= n.

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A344349 Number of primes along the main antidiagonal of the n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows.

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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.

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A342695 a(n) is the number of primes in an n X n square array that do not appear on its border, with the elements of the square array being the numbers from 1..n^2, listed in increasing order by rows.

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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.

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Examples

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Mathematica

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