A199333 Triangle read by rows: T(n,0) = T(n,n) = 1, 0 < k < n: T(n,k) = smallest prime not less than T(n-1,k) + T(n-1,k-1).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 7, 5, 1, 1, 7, 13, 13, 7, 1, 1, 11, 23, 29, 23, 11, 1, 1, 13, 37, 53, 53, 37, 13, 1, 1, 17, 53, 97, 107, 97, 53, 17, 1, 1, 19, 71, 151, 211, 211, 151, 71, 19, 1, 1, 23, 97, 223, 367, 431, 367, 223, 97, 23, 1, 1, 29, 127

Offset: 0

Author

Reinhard Zumkeller, Nov 09 2011

Keywords

• nonn
• tabl

Comments

T(n,k) = T(n,n-k);

T(n,0) = 1, cf. A000012;

T(n,1) = A008578(n), n > 0;

A199424(n) = first row in triangle A199302 containing n-th prime;

A199425(n) = number of distinct primes in rows 0 through n;

large terms in the b-file are probable primes only, row number > 50.

Examples

			0:                 1
1:               1   1
2:             1   2   1
3:           1   3   3   1
4:         1   5   7   5   1
5:       1   7  13  13   7   1
6:     1  11  23  29  23  11   1
7:   1  13  37  53  53  37  13   1
8: 1  17  53  97 107  97  53  17   1
primes in 8th row:
T(7,0) + T(7,1) = 1+13 = 14 --> T(8,1) = T(8,7) = 19;
T(7,1) + T(7,2) = 13+37 = 50 --> T(8,2) = T(8,6) = 53, already in row 7;
T(7,2) + T(7,3) = 37+53 = 90 --> T(8,3) = T(8,5) = 97;
T(7,3) + T(7,4) = 53+53 = 106 --> T(8,4) = 107.
		

Links

• Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A159477; A199581 & A199582 (central terms), A199694 (row sums), A199695 & A199696 (row products); A007318.

Cf. A132403, A362034.

Programs

Haskell

a199333 n k = a199333_tabl !! n !! k
a199333_row n = a199333_tabl !! n
a199333_list = concat a199333_tabl
a199333_tabl = iterate
   (\row -> map a159477 $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]

Mathematica

T[n_, k_] := T[n, k] = Switch[k, 0|n, 1, _, With[{m = T[n-1, k] + T[n-1, k-1]}, If[PrimeQ[m], m, NextPrime[m]]]];
Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

Formula

T(n,k) = A007918(T(n-1,k) + T(n-1,k-1)), 0 < k < n, T(n,0) = T(n,n) = 1.