OFFSET
1,4
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.
FORMULA
N_n# = product_{1<=j<=N, GCD(j, n) = 1, j is prime} j.
EXAMPLE
[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
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 *)
PROG
(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)]
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).
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Oct 02 2012
STATUS
approved