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

Author

Reinhard Zumkeller, Jan 19 2013

Keywords

• nonn

Comments

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

Examples

			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)
		

Links

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

Crossrefs

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

Programs

Haskell

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

Mathematica

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

PARI

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

Formula

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