A221490 -id:A221490 - cp's OEIS frontend

A209297 Triangle read by rows: T(n,k) = k*n + k - n, 1 <= k <= n.

1, 1, 4, 1, 5, 9, 1, 6, 11, 16, 1, 7, 13, 19, 25, 1, 8, 15, 22, 29, 36, 1, 9, 17, 25, 33, 41, 49, 1, 10, 19, 28, 37, 46, 55, 64, 1, 11, 21, 31, 41, 51, 61, 71, 81, 1, 12, 23, 34, 45, 56, 67, 78, 89, 100, 1, 13, 25, 37, 49, 61, 73, 85, 97, 109, 121, 1, 14, 27

Offset: 1

Reinhard Zumkeller, Jan 19 2013

From Michel Marcus, May 18 2021: (Start)
The n-th row of the triangle is 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.
                                             [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]
  -----------------------------------------------------------
   1       1 4      1 5 9     1  6  11 16     1  7  13 19 25
(End)

			From _Muniru A Asiru_, Oct 31 2017: (Start)
Triangle begins:
  1;
  1,  4;
  1,  5,  9;
  1,  6, 11, 16;
  1,  7, 13, 19, 25;
  1,  8, 15, 22, 29, 36;
  1,  9, 17, 25, 33, 41, 49;
  1, 10, 19, 28, 37, 46, 55, 64;
  1, 11, 21, 31, 41, 51, 61, 71, 81;
  1, 12, 23, 34, 45, 56, 67, 78, 89, 100;
  ... (End)

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A162610; A000012 (left edge), A000290 (right edge), A006003 (row sums), A001844 (central terms), A026741 (number of odd terms per row), A142150 (number of even terms per row), A221490 (number of primes per row).

Flat(List([1..10^3], n -> List([1..n], k -> k * n + k - n))); # Muniru A Asiru, Oct 31 2017

a209297 n k = k * n + k - n
a209297_row n = map (a209297 n) [1..n]
a209297_tabl = map a209297_row [1..]

Array[Range[1, #^2, #+1]&,10] (* Paolo Xausa, Feb 08 2024 *)

T(n,k) = (k-1)*(n+1)+1.

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)).

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.

A221491 Number of primes of the form k*n - k + n, 1 <= k <= n.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 5, 2, 3, 3, 5, 3, 6, 4, 7, 6, 5, 4, 9, 2, 8, 7, 8, 4, 11, 5, 10, 8, 10, 4, 16, 3, 10, 10, 12, 9, 15, 4, 10, 11, 13, 6, 20, 7, 11, 10, 15, 8, 19, 6, 19, 13, 16, 5, 25, 10, 19, 10, 15, 9, 26, 7, 16, 13, 16, 13, 30, 9, 18, 14, 26, 10, 25, 10

Offset: 1

Reinhard Zumkeller, Jan 19 2013

a(n) = sum(A010051(A162610(n,k)): k=1..n) = number of primes in n-th row of the triangle in A162610. - Reinhard Zumkeller, Jan 19 2013

			Row 10 of A162610 = [19,28,37,46,55,64,73,82,91,100] containing three primes: [19,37,73], therefore a(10) = 3;
row 11 of A162610 = [21,31,41,51,61,71,81,91,101,111,121] containing five primes: [31,41,61,71,101], therefore a(11) = 5.

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

Cf. A221490.

Cf. A010051, A162610.

a221491 n = sum [a010051' (k*n - k + n) | k <- [1..n]]

a[n_] := Sum[Boole[PrimeQ[(k+1)(n-1)+1]], {k, 1, n}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 20 2021 *)

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

A367077 Determinant of the n X n matrix whose terms are the n^2 values of isprime(x) from 1 to n^2.

Original entry on oeis.org

0, -1, -1, 0, 1, 0, -2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, -15, 0, 0, 0, 0, 0, 400, 0, -196, 0, 0, 0, 0, 0, 4224, 0, 0, 0, -44304, 0, -537138, 0, 0, 0, -4152330, 0, 0, 0, 0, 0, -59171526, 0, 0, 0, 0, 0, -1681340912, 0, 330218571840, 0, 0, 0, 0, 0, -349982854480, 0, 0, 0

Offset: 1

Andres Cicuttin, Nov 05 2023

sign

Traces of these matrices are A221490.

Consider the sequence b(n) defined as 0 when a(n) is 0 and 1 otherwise. What is the value of the limit as n approaches infinity of Sum_{j<=n} b(j)/n provided that this limit exists?

			For n=4, we consider the first n^2=16 terms of the characteristic function of primes (A010051): (0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0). These terms form a matrix by arranging them in 4 consecutive subsequences of 4 terms each:
  0, 1, 1, 0;
  1, 0, 1, 0;
  0, 0, 1, 0;
  1, 0, 0, 0;
and the determinant of this matrix is zero, so a(4)=0.

Cf. A010051, A064866, A289777, A367133, A221490.

mat[n_,i_,j_]:=Boole[PrimeQ[(i-1)*n+j]];
a[n_]:=Det@Table[mat[n,i,j],{i,1,n},{j,1,n}];
Table[a[n],{n,1,70}]

a(n) = matdet(matrix(n, n, i, j, isprime((i-1)*n+j))); \\ Michel Marcus, Nov 07 2023

from sympy import Matrix, isprime
def A367077(n): return Matrix(n,n,[int(isprime(i)) for i in range(1,n**2+1)]).det() # Chai Wah Wu, Nov 16 2023

A367133 Rank of the n X n matrix whose terms are the n^2 values of isprime(x) from 1 to n^2.

Original entry on oeis.org

0, 2, 3, 3, 5, 3, 7, 5, 7, 5, 11, 5, 11, 7, 9, 9, 16, 7, 19, 9, 13, 11, 23, 9, 20, 13, 19, 13, 29, 9, 31, 17, 21, 17, 25, 13, 37, 19, 25, 17, 41, 13, 43, 21, 25, 23, 47, 17, 43, 21, 33, 25, 53, 19, 41, 25, 37, 29, 59, 17, 61, 31, 37, 33, 49, 21, 67, 33, 45, 25

Offset: 1

Andres Cicuttin, Nov 06 2023

nonn

Traces of these matrices are A221490.

			For n=4, we consider the first n^2=16 terms of the characteristic function of primes (A010051): (0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0). These terms form a matrix by arranging them in 4 consecutive subsequences of 4 terms each:
  0, 1, 1, 0;
  1, 0, 1, 0;
  0, 0, 1, 0;
  1, 0, 0, 0;
and the largest square submatrix with a nonzero determinant within this matrix is of dimension 3. Therefore, the rank is 3, and so a(4)=3.

Cf. A010051, A064866, A289777, A367077, A221490.

f:= proc(n) local i; LinearAlgebra:-Rank(Matrix(n,n,[seq(`if`(isprime(i),1,0),i=1..n^2)])) end proc:
map(f, [$1..100]); # Robert Israel, Nov 11 2023

mat[n_,i_,j_]:=Boole[PrimeQ[(i-1)*n+j]];
a[n_]:=MatrixRank@Table[mat[n,i,j],{i,1,n},{j,1,n}];
Table[a[n],{n,1,70}]

a(n) = matrank(matrix(n, n, i, j, isprime((i-1)*n+j))); \\ Michel Marcus, Nov 07 2023

from sympy import Matrix, isprime
def A367133(n): return Matrix(n,n,[int(isprime(i)) for i in range(1,n**2+1)]).rank() # Chai Wah Wu, Nov 16 2023

A350328 Numbers k such that pi(k) = Sum_{i=1..k} pi(k(i-1)+i) - pi(k(i-1)+i-1).

Original entry on oeis.org

1, 5, 21, 23, 25, 33, 81, 85, 115, 127, 141, 164, 253, 273, 283, 285, 291, 343, 385, 441, 471, 495, 505, 565, 577, 711, 807, 921, 1107, 1977, 2175, 2437, 2941, 2943, 3381, 4117, 5541, 6531, 7075, 7497, 8193, 8325, 8923

Offset: 1

Wesley Ivan Hurt, Dec 24 2021

nonn

Numbers with the same number of primes on the top row and 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 example).

			5 is in the sequence since there are 3 primes in the top row and 3 primes along the main diagonal 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.

Numbers k such that A000720(k) = A221490(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

A209297 Triangle read by rows: T(n,k) = 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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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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A221491 Number of primes of the form k*n - k + n, 1 <= k <= n.

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A367077 Determinant of the n X n matrix whose terms are the n^2 values of isprime(x) from 1 to n^2.

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A367133 Rank of the n X n matrix whose terms are the n^2 values of isprime(x) from 1 to n^2.

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A350328 Numbers k such that pi(k) = Sum_{i=1..k} pi(k*(i-1)+i) - pi(k*(i-1)+i-1).

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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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A350328 Numbers k such that pi(k) = Sum_{i=1..k} pi(k(i-1)+i) - pi(k(i-1)+i-1).

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)).