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

%I A216914 #17 Oct 20 2019 02:01:30
%S A216914 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 A216914 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 A216914 70,105,6,5,6,3,2,1,1,2310,105,70,105,42,5,30,3,2
%N A216914 The Gauss factorial N_n! restricted to prime factors for N >= 0, n >= 1, square array read by antidiagonals.
%C A216914 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.
%C A216914 Following the style of A034386 we will write N_n# for A(N,n) and call N_n# the Gauss primorial.
%F A216914 N_n# = product_{1<=j<=N, GCD(j, n) = 1, j is prime} j.
%e A216914 [n\N][0, 1, 2, 3, 4,  5,  6,   7,   8,   9, 10]
%e A216914 -----------------------------------------------
%e A216914 [ 1]  1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210
%e A216914 [ 2]  1, 1, 1, 3, 3, 15, 15, 105, 105, 105, 105
%e A216914 [ 3]  1, 1, 2, 2, 2, 10, 10,  70,  70,  70,  70
%e A216914 [ 4]  1, 1, 1, 3, 3, 15, 15, 105, 105, 105, 105
%e A216914 [ 5]  1, 1, 2, 6, 6,  6,  6,  42,  42,  42,  42
%e A216914 [ 6]  1, 1, 1, 1, 1,  5,  5,  35,  35,  35,  35
%e A216914 [ 7]  1, 1, 2, 6, 6, 30, 30,  30,  30,  30,  30
%e A216914 [ 8]  1, 1, 1, 3, 3, 15, 15, 105, 105, 105, 105
%e A216914 [ 9]  1, 1, 2, 2, 2, 10, 10,  70,  70,  70,  70
%e A216914 [10]  1, 1, 1, 3, 3,  3,  3,  21,  21,  21,  21
%e A216914 [11]  1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210
%e A216914 [12]  1, 1, 1, 1, 1,  5,  5,  35,  35,  35,  35
%e A216914 [13]  1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210
%t A216914 (* k stands for N *) T[n_, k_] := Product[If[GCD[j, n] == 1 && PrimeQ[j], j, 1], {j, 1, k}];
%t A216914 Table[T[n - k, k], {n, 1, 12}, {k, n - 1, 0, -1}] // Flatten (* _Jean-François Alcover_, Aug 02 2019 *)
%o A216914 (Sage)
%o A216914 def Gauss_primorial(N, n):
%o A216914     return mul(j for j in (1..N) if gcd(j, n) == 1 and is_prime(j))
%o A216914 for n in (1..13): [Gauss_primorial(N,n) for N in (1..10)]
%Y A216914 Cf. A034386(n) = n# = Gauss_primorial(n, 1).
%Y A216914 The compressed version of the primorial eliminates all duplicates.
%Y A216914 Cf. A002110(n) = compressed(Gauss_primorial(n, 1)).
%Y A216914 Cf. A070826(n) = compressed(Gauss_primorial(n, 2)).
%Y A216914 Cf. A007947(n) = Gauss_primorial(1*n, 1)/Gauss_primorial(1*n, 1*n).
%Y A216914 Cf. A204455(n) = Gauss_primorial(2*n, 2)/Gauss_primorial(2*n, 2*n).
%Y A216914 Cf. A216913(n) = Gauss_primorial(3*n, 3)/Gauss_primorial(3*n, 3*n).
%K A216914 nonn,tabl
%O A216914 1,4
%A A216914 _Peter Luschny_, Oct 02 2012