A199696 -id:A199696 - 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.

A199582 Central terms of the triangle in A199333: a(n) = A199333(n,[n/2]).

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 29, 53, 107, 211, 431, 809, 1619, 3037, 6079, 11467, 22937, 43541, 87083, 166183, 332393, 636761, 1273541, 2448049, 4896103, 9438851, 18877711, 36484271, 72968563, 141332173, 282664351, 548544487, 1097088989, 2132671027, 4265342057

Offset: 0

Reinhard Zumkeller, Nov 09 2011

nonn

a(n) = max(A199333(n,k): 0<=k<=n/2) = A006530(A199695(n)) = A006530(A199696(n));

a(2*n) = A007918(a(2*n-1)) for n > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 0..150

Cf. A199581, A001405.

a199582 n = a199333_row n !! (n `div` 2)

A199695 Row products of the triangle in A199333.

Original entry on oeis.org

1, 1, 2, 9, 175, 8281, 1856261, 649893049, 817291210163, 1847322434679121, 14368726069959027071, 342031303262647675287601, 13964481217238950868653586531, 1889891784470148590323094656731121, 586215019967842464352819482405063771511

Offset: 0

Reinhard Zumkeller, Nov 09 2011

nonn

a(n) = Product_{k=0..n} A199333(n,k);
A199696(n) = A007947(a(n));
A020639(a(n)) = A008578(n); A006530(a(n)) = A199582(n).

Cf. A199333, A199694.

Haskell
```
a199695 = product . a199333_row
```

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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A199582 Central terms of the triangle in A199333: a(n) = A199333(n,[n/2]).

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A199695 Row products of the triangle in A199333.

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