A216914 The Gauss factorial N_n! restricted to prime factors for N >= 0, n >= 1, square array read by antidiagonals.

1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 6, 3, 2, 1, 1, 30, 3, 2, 1, 1, 1, 30, 15, 2, 3, 2, 1, 1, 210, 15, 10, 3, 6, 1, 1, 1, 210, 105, 10, 15, 6, 1, 2, 1, 1, 210, 105, 70, 15, 6, 1, 6, 1, 1, 1, 210, 105, 70, 105, 6, 5, 6, 3, 2, 1, 1, 2310, 105, 70, 105, 42, 5, 30, 3, 2

Offset: 1

Author

Peter Luschny, Oct 02 2012

Keywords

• nonn
• tabl

Comments

The term Gauss factorial N_n! was introduced by J. B. Cosgrave and K. Dilcher (see references in A216919). It is closely related to the Gauss-Wilson theorem which was stated in Gauss' Disquisitiones Arithmeticae (§78). Restricting the factors of the Gauss factorial to primes gives the present sequence.

Following the style of A034386 we will write N_n# for A(N,n) and call N_n# the Gauss primorial.

Examples

			[n\N][0, 1, 2, 3, 4,  5,  6,   7,   8,   9, 10]
-----------------------------------------------
[ 1]  1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210
[ 2]  1, 1, 1, 3, 3, 15, 15, 105, 105, 105, 105
[ 3]  1, 1, 2, 2, 2, 10, 10,  70,  70,  70,  70
[ 4]  1, 1, 1, 3, 3, 15, 15, 105, 105, 105, 105
[ 5]  1, 1, 2, 6, 6,  6,  6,  42,  42,  42,  42
[ 6]  1, 1, 1, 1, 1,  5,  5,  35,  35,  35,  35
[ 7]  1, 1, 2, 6, 6, 30, 30,  30,  30,  30,  30
[ 8]  1, 1, 1, 3, 3, 15, 15, 105, 105, 105, 105
[ 9]  1, 1, 2, 2, 2, 10, 10,  70,  70,  70,  70
[10]  1, 1, 1, 3, 3,  3,  3,  21,  21,  21,  21
[11]  1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210
[12]  1, 1, 1, 1, 1,  5,  5,  35,  35,  35,  35
[13]  1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210
		

Crossrefs

Cf. A034386(n) = n# = Gauss_primorial(n, 1).

The compressed version of the primorial eliminates all duplicates.

Cf. A002110(n) = compressed(Gauss_primorial(n, 1)).

Cf. A070826(n) = compressed(Gauss_primorial(n, 2)).

Cf. A007947(n) = Gauss_primorial(1*n, 1)/Gauss_primorial(1*n, 1*n).

Cf. A204455(n) = Gauss_primorial(2*n, 2)/Gauss_primorial(2*n, 2*n).

Cf. A216913(n) = Gauss_primorial(3*n, 3)/Gauss_primorial(3*n, 3*n).

Programs

Mathematica

(* k stands for N *) T[n_, k_] := Product[If[GCD[j, n] == 1 && PrimeQ[j], j, 1], {j, 1, k}];
Table[T[n - k, k], {n, 1, 12}, {k, n - 1, 0, -1}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

Sage

def Gauss_primorial(N, n):
    return mul(j for j in (1..N) if gcd(j, n) == 1 and is_prime(j))
for n in (1..13): [Gauss_primorial(N,n) for N in (1..10)]

Formula

N_n# = product_{1<=j<=N, GCD(j, n) = 1, j is prime} j.