A344349 -id:A344349 - 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

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.

Original entry on oeis.org

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

Wesley Ivan Hurt, May 14 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            4                 5
------------------------------------------------------------------------
  primes   {}      {2,3}    {2,3,7}    {2,3,5,13}       {2,3,5,11,23}
------------------------------------------------------------------------

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

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

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

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)).

A350377 Numbers k such that Sum_{j=1..k} (pi(kj-j+1) - pi(kj-j)) = Sum_{i=1..k} (pi(k(i-1)+i) - pi(k(i-1)+i-1)).

Original entry on oeis.org

1, 5, 8, 10, 11, 12, 14, 21, 23, 24, 27, 63, 64, 72, 90, 99, 144, 176, 184, 340, 366, 393, 480, 567, 693, 915, 975, 1046, 1068, 1084, 1260, 1410, 1452, 1830, 1968, 2268, 2490, 2943, 3087, 3735, 5284, 5426, 5637, 5757, 6015, 6334, 6393, 6570, 6582, 8292, 9836, 10005

Offset: 1

Wesley Ivan Hurt, Dec 28 2021

nonn

Numbers with the same number of primes appearing along the main diagonal and along the main antidiagonal of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows (see example).

			5 is in the sequence since there are 3 primes along the main diagonal and 3 primes along the main antidiagonal of the 5 X 5 array below.
  [1   2  3  4  5]
  [6   7  8  9 10]
  [11 12 13 14 15]
  [16 17 18 19 20]
  [21 22 23 24 25]

Cf. A000720 (pi), A221490, A344349, A350328.

q[k_] := Sum[Boole @ PrimeQ[k*j - j + 1] - Boole @ PrimeQ[k*(j - 1) + j], {j, 1, k}] == 0; Select[Range[1000], q] (* Amiram Eldar, Dec 28 2021 *)

Numbers k such that A221490(k) = A344349(k).

More terms from Amiram Eldar, Dec 28 2021

cp's OEIS Frontend

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

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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.

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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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A350377 Numbers k such that Sum_{j=1..k} (pi(k*j-j+1) - pi(k*j-j)) = Sum_{i=1..k} (pi(k*(i-1)+i) - pi(k*(i-1)+i-1)).

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A350377 Numbers k such that Sum_{j=1..k} (pi(kj-j+1) - pi(kj-j)) = Sum_{i=1..k} (pi(k(i-1)+i) - pi(k(i-1)+i-1)).