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A216914 The Gauss factorial N_n! restricted to prime factors for N >= 0, n >= 1, square array read by antidiagonals. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..75.

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).

Sequence in context: A321716 A245567 A204168 * A216917 A216919 A152656

Adjacent sequences:  A216911 A216912 A216913 * A216915 A216916 A216917

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Oct 02 2012

STATUS

approved

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Last modified May 25 08:35 EDT 2020. Contains 334585 sequences. (Running on oeis4.)