A096334 Triangle read by rows: T(n,k) = prime(n)#/prime(k)#, 0<=k<=n.

1, 2, 1, 6, 3, 1, 30, 15, 5, 1, 210, 105, 35, 7, 1, 2310, 1155, 385, 77, 11, 1, 30030, 15015, 5005, 1001, 143, 13, 1, 510510, 255255, 85085, 17017, 2431, 221, 17, 1, 9699690, 4849845, 1616615, 323323, 46189, 4199, 323, 19, 1, 223092870, 111546435, 37182145, 7436429, 1062347, 96577, 7429, 437, 23, 1

Offset: 0

Author

Reinhard Zumkeller, Aug 03 2004

Keywords

• nonn
• tabl

Comments

T(n,k) is the (k+1)-th product of (n-k) successive primes (k, n-(k+1) >= 0). - Alois P. Heinz, Jan 21 2022

Examples

			Triangle begins:
    1;
    2,   1;
    6,   3,  1;
   30,  15,  5, 1;
  210, 105, 35, 7, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

T(n-1+j,n-1) give (j=1-12): A000040, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127342, A127343, A127344.

Columns k=0-1 give: A002110, A070826.

T(2n,n) gives A107712.

Row sums give A350895.

Antidiagonal sums give A350758.

Cf. A073485 (distinct values sorted).

Programs

Maple

T:= proc(n, k) option remember;
     `if`(n=k, 1, T(n-1, k)*ithprime(n))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 21 2022

Mathematica

T[n_, k_] := Times @@ Prime[Range[k + 1, n]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 13 2021 *)

PARI

pr(n) = factorback(primes(n)); \\ A002110
row(n) = my(P=pr(n)); vector(n+1, k, P/pr(k-1)); \\ Michel Marcus, Jan 21 2022

Formula

T(n,0) = A002110(n); T(n,n) = 1;

T(n,n-1) = A000040(n) for n>0;

T(n,k) = A002110(n)/A002110(k), 0<=k<=n.

T(n,k) = Product_{j=k+1..n} prime(j). - Alois P. Heinz, Jan 21 2022