%I
%S 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,
%T 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,
%U 70,105,6,5,6,3,2,1,1,2310,105,70,105,42,5,30,3,2
%N The Gauss factorial N_n! restricted to prime factors for N >= 0, n >= 1, square array read by antidiagonals.
%C 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 GaussWilson theorem which was stated in Gauss' Disquisitiones Arithmeticae (§78). Restricting the factors of the Gauss factorial to primes gives the present sequence.
%C Following the style of A034386 we will write N_n# for A(N,n) and call N_n# the Gauss primorial.
%F N_n# = product_{1<=j<=N, GCD(j, n) = 1, j is prime} j.
%e [n\N][0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
%e 
%e [ 1] 1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210
%e [ 2] 1, 1, 1, 3, 3, 15, 15, 105, 105, 105, 105
%e [ 3] 1, 1, 2, 2, 2, 10, 10, 70, 70, 70, 70
%e [ 4] 1, 1, 1, 3, 3, 15, 15, 105, 105, 105, 105
%e [ 5] 1, 1, 2, 6, 6, 6, 6, 42, 42, 42, 42
%e [ 6] 1, 1, 1, 1, 1, 5, 5, 35, 35, 35, 35
%e [ 7] 1, 1, 2, 6, 6, 30, 30, 30, 30, 30, 30
%e [ 8] 1, 1, 1, 3, 3, 15, 15, 105, 105, 105, 105
%e [ 9] 1, 1, 2, 2, 2, 10, 10, 70, 70, 70, 70
%e [10] 1, 1, 1, 3, 3, 3, 3, 21, 21, 21, 21
%e [11] 1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210
%e [12] 1, 1, 1, 1, 1, 5, 5, 35, 35, 35, 35
%e [13] 1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210
%t (* k stands for N *) T[n_, k_] := Product[If[GCD[j, n] == 1 && PrimeQ[j], j, 1], {j, 1, k}];
%t Table[T[n  k, k], {n, 1, 12}, {k, n  1, 0, 1}] // Flatten (* _JeanFrançois Alcover_, Aug 02 2019 *)
%o (Sage)
%o def Gauss_primorial(N, n):
%o return mul(j for j in (1..N) if gcd(j, n) == 1 and is_prime(j))
%o for n in (1..13): [Gauss_primorial(N,n) for N in (1..10)]
%Y Cf. A034386(n) = n# = Gauss_primorial(n, 1).
%Y The compressed version of the primorial eliminates all duplicates.
%Y Cf. A002110(n) = compressed(Gauss_primorial(n, 1)).
%Y Cf. A070826(n) = compressed(Gauss_primorial(n, 2)).
%Y Cf. A007947(n) = Gauss_primorial(1*n, 1)/Gauss_primorial(1*n, 1*n).
%Y Cf. A204455(n) = Gauss_primorial(2*n, 2)/Gauss_primorial(2*n, 2*n).
%Y Cf. A216913(n) = Gauss_primorial(3*n, 3)/Gauss_primorial(3*n, 3*n).
%K nonn,tabl
%O 1,4
%A _Peter Luschny_, Oct 02 2012
