A216914 - OEIS

%I #17 Oct 20 2019 02:01:30

%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 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 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 (* _Jean-François 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