A221491 -id:A221491 - cp's OEIS frontend

A162610 Triangle read by rows in which row n lists n terms, starting with 2n-1, with gaps = n-1 between successive terms.

1, 3, 4, 5, 7, 9, 7, 10, 13, 16, 9, 13, 17, 21, 25, 11, 16, 21, 26, 31, 36, 13, 19, 25, 31, 37, 43, 49, 15, 22, 29, 36, 43, 50, 57, 64, 17, 25, 33, 41, 49, 57, 65, 73, 81, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121

Offset: 1

Omar E. Pol, Jul 09 2009

Note that the last term of the n-th row is the n-th square A000290(n).

Row sums are n*(n^2+2*n-1)/2, apparently in A127736. - R. J. Mathar, Jul 20 2009

			Triangle begins:
1
3, 4
5, 7, 9
7, 10, 13, 16
9, 13, 17, 21, 25
11, 16, 21, 26, 31, 36

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A000027, A000290, A159797, A159798.

Cf. A209297; A005408 (left edge), A000290 (right edge), A127736 (row sums), A056220 (central terms), A026741 (number of odd terms per row), A142150 (number of even terms per row), A221491 (number of primes per row).

a162610 n k = k * n - k + n
a162610_row n = map (a162610 n) [1..n]
a162610_tabl = map a162610_row [1..]
-- Reinhard Zumkeller, Jan 19 2013

Flatten[Table[NestList[#+n-1&,2n-1,n-1], {n,15}]] (* Harvey P. Dale, Oct 20 2011 *)

# From R. J. Mathar, Oct 20 2009
def A162610(n, k):
    return 2*n-1+(k-1)*(n-1)
print([A162610(n,k) for n in range(1,20) for k in range(1,n+1)])

T(n,k) = n+k*n-k, 1<=k<=n. - R. J. Mathar, Oct 20 2009

T(n,k) = (k+1)*(n-1)+1. - Reinhard Zumkeller, Jan 19 2013

More terms from R. J. Mathar, Oct 20 2009

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

Original entry on oeis.org

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

cp's OEIS Frontend

A162610 Triangle read by rows in which row n lists n terms, starting with 2n-1, with gaps = n-1 between successive terms.

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