A127344 -id:A127344 - cp's OEIS frontend

A127343 Product of 11 consecutive primes.

200560490130, 3710369067405, 50708377254535, 436092044389001, 2928046583754721, 14107860812636383, 64027983688118969, 229747470880897477, 810162134158954261, 2500935283708076197

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) is the absolute value of the coefficient of x^0 of the polynomial Product_{j=0..10} (x-prime(n+j)) of degree 11; the roots of this polynomial are prime(n), ..., prime(n+10).

Cf. A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127342, A127344.

[&*[ NthPrime(n+k): k in [0..10] ]: n in [1..50] ]; // Vincenzo Librandi, Apr 03 2011

a = {}; Do[AppendTo[a, Product[Prime[x + n], {n, 0, 10}]], {x, 1, 50}]; a
Times@@@Partition[Prime[Range[50]],11,1] (* Harvey P. Dale, Oct 21 2011 *)

{m=10;k=11;for(n=0,m-1,print1(a=prod(j=1,k,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=10;k=11;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),0)),","))} \\ Klaus Brockhaus, Jan 21 2007

Edited by Klaus Brockhaus, Jan 21 2007

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

Original entry on oeis.org

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

Reinhard Zumkeller, Aug 03 2004

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

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

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

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).

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

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 *)

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

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

A127342 Product of 10 consecutive primes.

Original entry on oeis.org

6469693230, 100280245065, 1236789689135, 10141675450907, 62298863484143, 266186053068611, 1085220062510491, 3766351981654057, 12091972151626183, 35224440615606707, 86239147714071593, 203079283326684719

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) = coefficient of x^0 of the polynomial Product_{j=0..9} (x-prime(n+j)) of degree 10; the roots of this polynomial are prime(n), ..., prime(n+9).

Cf. A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127343, A127344.

[&*[ NthPrime(n+k): k in [0..9] ]: n in [1..50] ]; // Vincenzo Librandi, Apr 03 2011

a = {}; Do[AppendTo[a, Product[Prime[x + n], {n, 0, 9}]], {x, 1, 50}]; a
Times@@@Partition[Prime[Range[50]],10,1] (* Harvey P. Dale, Oct 21 2011 *)

{m=12;k=10;for(n=0,m-1,print1(a=prod(j=1,k,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=12;k=10;for(n=1,m,print1(polcoeff(prod(j=0,k-1,(x-prime(n+j))),0),","))} \\ Klaus Brockhaus, Jan 21 2007

Edited by Klaus Brockhaus, Jan 21 2007

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A127343 Product of 11 consecutive primes.

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

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A127342 Product of 10 consecutive primes.

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