A132403 -id:A132403 - cp's OEIS frontend

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

Reinhard Zumkeller, Nov 09 2011

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.

			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.

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

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

Cf. A132403, A362034.

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]

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 *)

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

A362034 Triangle read by rows: T(n,0) = T(n,n) = 2, 0 < k < n: T(n,k) = smallest prime not less than T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

2, 2, 2, 2, 5, 2, 2, 7, 7, 2, 2, 11, 17, 11, 2, 2, 13, 29, 29, 13, 2, 2, 17, 43, 59, 43, 17, 2, 2, 19, 61, 103, 103, 61, 19, 2, 2, 23, 83, 167, 211, 167, 83, 23, 2, 2, 29, 107, 251, 379, 379, 251, 107, 29, 2, 2, 31, 137, 359, 631, 761, 631, 359, 137, 31, 2

Offset: 0

Jack Braxton, Apr 05 2023

In order to get the next number in the row, you add the two numbers above it, and find the next prime.
3 is the only prime number that never shows up.
5 is the only prime number that only shows up once; every prime number above 5 shows up at least twice.

			Triangle begins:
      k=0  1   2   3   4   5   6   7   8  9 10
  n=0:  2
  n=1:  2  2
  n=2:  2  5   2
  n=3:  2  7   7   2
  n=4:  2 11  17  11   2
  n=5:  2 13  29  29  13   2
  n=6:  2 17  43  59  43  17   2
  n=7:  2 19  61 103 103  61  19   2
  n=8:  2 23  83 167 211 167  83  23   2
  n=9:  2 29 107 251 379 379 251 107  29  2
 n=10:  2 31 137 359 631 761 631 359 137 31  2

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0..150, flattened)

Cf. A000040, A007318, A132403, A199333.

for n from 0 to 10 do
  T[n,0]:= 2: T[n,n]:= 2:
  for k from 1 to n-1 do
    T[n,k]:= nextprime(T[n-1,k-1]+T[n-1,k]-1)
  od
od:
for n from 0 to 10 do
  seq(T[n,k],k=0..n)
od; # Robert Israel, Apr 05 2023

T[n_, 0] := T[n, n] = 2; T[n_, k_] := T[n, k] = NextPrime[T[n - 1, k - 1] + T[n - 1, k] - 1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2023, after Maple *)

T(n,k) = if (n==0, 2, if (k==0, 2, if (k==n, 2, nextprime(T(n-1,k-1) + T(n-1,k))))); \\ Michel Marcus, Apr 07 2023

T(n,k) = A007918(T(n-1,k-1) + T(n-1,k)) for 0 < k < n. - Robert Israel, Apr 05 2023

cp's OEIS Frontend

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).

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