A216919 -id:A216919 - cp's OEIS frontend

A001783 n-phi-torial, or phi-torial of n: Product k, 1 <= k <= n, k relatively prime to n.

1, 1, 2, 3, 24, 5, 720, 105, 2240, 189, 3628800, 385, 479001600, 19305, 896896, 2027025, 20922789888000, 85085, 6402373705728000, 8729721, 47297536000, 1249937325, 1124000727777607680000, 37182145, 41363226782215962624, 608142583125, 1524503639859200000

Offset: 1

N. J. A. Sloane

In other words, a(1) = 1 and for n >= 2, a(n) = product of the phi(n) numbers < n and relatively prime to n.
From Gauss's generalization of Wilson's theorem (see Weisstein link) it follows that, for n>1, a(n) == -1 (mod n) if and only if there exists a primitive root modulo n (cf. the Hardy and Wright reference, Theorem 129. p. 102). - Vladimir Shevelev, May 11 2012
Islam & Manzoor prove that a(n) is an injection for n > 1, see links. In other words, if a(m) = a(n), and min(m, n) > 1, then m = n. - Muhammed Hedayet, May 25 2016
Cosgrave & Dilcher propose the name Gauss factorial. Indeed the sequence is the special case N = n of the Gauss factorial N_n! = Product_{1<=j<=N, gcd(j, n)=1} j (see A216919). - Peter Luschny, Feb 07 2018

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 129, p. 102.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..200
J. B. Cosgrave and K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008)
J. B. Cosgrave and K. Dilcher, The multiplicative orders of certain Gauss factorials, Intl. J. Number Theory 7 (1) (2011) 145-171.
S. W. Golomb and William Small, Problem E1045, Amer. Math. Monthly, 60 (1953), 422.
Muhammed H. Islam and Shahriar Manzoor, φ1 and phitorial are injections, for any positive integer N, where N > 1
Laszlo Toth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7
Eric Weisstein's World of Mathematics, Wilson's Theorem

Cf. A023896, A066570, A038566, A124441.

Main diagonal gives A216919.

a001783 = product . a038566_row
-- Reinhard Zumkeller, Mar 04 2012, Aug 26 2011

A001783 := proc(n) local i,t1; t1 := 1; for i from 1 to n do if gcd(i,n)=1 then t1 := t1*i; fi; od; t1; end;
A001783 := proc(n) local i; mul(i,i=select(k->igcd(n,k)=1,[$1..n])) end; # Peter Luschny, Oct 30 2010

A001783[n_]:=Times@@Select[Range[n],CoprimeQ[n,#]&];
Array[A001783,20] (* Enrique Pérez Herrero, Jul 23 2011 *)

A001783(n)=prod(k=2,n-1,k^(gcd(k,n)==1))  \\ M. F. Hasler, Jul 23 2011

a(n)=my(f=factor(n),t=n^eulerphi(f)); fordiv(f,d, t*=(d!/d^d)^moebius(n/d)); t \\ Charles R Greathouse IV, Nov 05 2015

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A001783(n): return Gauss_factorial(n, n)
[A001783(n) for n in (1..25)] # Peter Luschny, Oct 01 2012

a(n) = n^phi(n)*Product_{d|n} (d!/d^d)^mu(n/d); phi=A000010 is the Euler totient function and mu=A008683 the Moebius function (Tom M. Apostol, Introduction to Analytic Number Theory, New York 1984, p. 48). - Franz Vrabec, Jul 08 2005
a(n) = n!/A066570(n). - R. J. Mathar, Mar 10 2011
A001221(a(n)) = A000720(n) - A001221(n) = A048865(n).
A006530(a(n)) = A136548(n). - Enrique Pérez Herrero, Jul 23 2011
a(n) = A124441(n)*A124442(n). - M. F. Hasler, Jul 23 2011
a(n) == (-1)^A211487(n) (mod n). - Vladimir Shevelev, May 13 2012
a(n) = A250269(n) / A193679(n). - Daniel Suteu, Apr 05 2021

More terms from James Sellers, Dec 23 1999

A066570 Product of numbers <= n that have a prime factor in common with n.

Original entry on oeis.org

1, 2, 3, 8, 5, 144, 7, 384, 162, 19200, 11, 1244160, 13, 4515840, 1458000, 10321920, 17, 75246796800, 19, 278691840000, 1080203040, 899245670400, 23, 16686729658368000, 375000, 663152807116800, 7142567040, 209964381084057600, 29, 1229978843118305280000000

Offset: 1

Amarnath Murthy, Dec 19 2001

Empty product, 1, for n = 1.

a(p) = p if p is a prime.

			a(7) = 7, a(9) = 3*6*9 = 162.

T. D. Noe, Table of n, a(n) for n = 1..200

Cf. A001783, A216919.

A066570 := proc(n) local i; mul(i,i=remove(k->igcd(n,k)=1,[$1..n])) end: # Peter Luschny, Oct 11 2011

Table[Times @@ Select[Range[2, n], GCD[#, n] > 1 &], {n, 30}] (* T. D. Noe, Oct 04 2012 *)

a(n) = prod(k=1, n, if (gcd(k, n) != 1, k, 1)); \\ Michel Marcus, Nov 02 2017

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A066570(n): return Gauss_factorial(n, 1)/Gauss_factorial(n, n)
[A066570(n) for n in (1..30)] # Peter Luschny, Oct 02 2012

a(n) = n!/A001783(n).

a(n) = Gauss_factorial(n, 1)/Gauss_factorial(n, n) (see A216919). - Peter Luschny, Oct 02 2012

A103131 The product of the residues in [1,n] relatively prime to n, taken modulo n.

Original entry on oeis.org

0, 1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, -1, 1, 1, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1

Offset: 1

Eric W. Weisstein, Jan 23 2005

sign

Old name was: Minimal residue (in absolute value) of A001783(n) (mod n).
If the positive representation for integers modulo n is used this is A160377. Here the symmetric (or minimal) representation for the integers modulo n is used.
From Gauss's generalization of Wilson's theorem (see Weisstein link) it follows that, for n>1, a(n) = -1 if and only if there exists a primitive root modulo n (cf. the Hardy and Wright reference, Theorem 129. p. 102). (Adapted from a comment by Vladimir Shevelev in A001783). - Peter Luschny, Oct 20 2012

			The residues in [1, 22] relatively prime to 22 are [1, 3, 5, 7, 9, 13, 15, 17, 19, 21] and the product of those residues is -1 modulo 22.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 129, p. 102.

Antti Karttunen, Table of n, a(n) for n = 1..16385
J. B. Cosgrave and K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008).
Eric Weisstein's World of Mathematics, Wilson's theorem

Cf. A001783, A160377, A211487, A216919.

A103131 := proc(n) local k, r; r := 1;
for k to n do if igcd(n,k) = 1 then r := mods(r*k, n) fi od;
r end: seq(A103131(i), i=1..102);   # Peter Luschny, Oct 20 2012

a[n_] := If[IntegerQ[PrimitiveRoot[n]], -1, 1]; a[1] = 0; a[2] = 1; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Nov 09 2012, after Max Alekseyev *)

A211487(n) = if(n%2, !!isprimepower(n), (n==2 || n==4 || (isprimepower(n/2, &n) && n>2))); \\ After Charles R Greathouse IV's code for A033948.
A103131(n) = if(n<=2,n-1,(-1)^A211487(n)); \\ Antti Karttunen, Aug 22 2017

def A103131(n):
    def smod(a,n): return a-n*ceil(a/n-1/2) if n != 0 else a
    r = 1
    for k in (1..n):
        if gcd(n, k) == 1: r = smod(r*k, n)
    return r
[A103131(n) for n in (1..102)]  # Peter Luschny, Oct 20 2012

For n>2, a(n)=-1 if A060594(n)=2, or equivalently if n is in A033948; otherwise a(n)=1. - Max Alekseyev, May 26 2009; edited by Peter Luschny, May 25 2017.
a(n) = Gauss_factorial(n, n) modulo n. (Definition of the Gauss factorial in A216919.) - Peter Luschny, Oct 20 2012
For n > 2, a(n) = (-1)^A211487(n). (See Max Alekseyev's formula above.) - Antti Karttunen, Aug 22 2017

Definition rewritten by Max Alekseyev, May 26 2009
New name from Peter Luschny, Oct 20 2012
a(2) set to 1 by Peter Luschny, May 25 2017

A160377 Phi-torial of n (A001783) modulo n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 1, 12, 13, 1, 1, 16, 17, 18, 1, 1, 21, 22, 1, 24, 25, 26, 1, 28, 1, 30, 1, 1, 33, 1, 1, 36, 37, 1, 1, 40, 1, 42, 1, 1, 45, 46, 1, 48, 49, 1, 1, 52, 53, 1, 1, 1, 57, 58, 1, 60, 61, 1, 1, 1, 1, 66, 1, 1, 1, 70, 1, 72, 73, 1, 1, 1, 1, 78, 1, 80, 81, 82, 1, 1, 85, 1, 1

Offset: 1

J. M. Bergot, May 11 2009

nonn

Is a(n)<> 1 iff n in A033948, n>2? [R. J. Mathar, May 21 2009]

Same as A103131, except there -1 appears instead of n-1. By Gauss's generalization of Wilson's theorem, a(n)=-1 means n has a primitive root (n in A033948) and a(n)=1 means n has no primitive root (n in A033949). [T. D. Noe, May 21 2009]

			Phi-torial of 12 equals 1*5*7*11=385 which leaves a remainder of 1 when divided by 12.
Phi-torial of 14 equals 1*3*5*9*11*13=19305 which leaves a remainder of 13 when divided by 14.

Amiram Eldar, Table of n, a(n) for n = 1..10000
John B. Cosgrave and Karl Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 8 (2008), Article #A39.
Eric Weisstein's World of Mathematics, Wilson's Theorem.

Cf. A124740 (one of just four listing "product of coprimes").

copr := proc(n) local a,k ; a := {1} ; for k from 2 to n-1 do if gcd(k,n) = 1 then a := a union {k} ; fi; od: a ; end:
A001783 := proc(n) local c; mul(c,c= copr(n)) ; end:
A160377 := proc(n) A001783(n) mod n ; end: seq( A160377(n),n=1..100) ; # R. J. Mathar, May 21 2009
A160377 := proc(n) local k, r; r := 1:
for k to n do if igcd(n,k) = 1 then r := modp(r*k, n) fi od;
r end: seq( A160377(i), i=1..88); # Peter Luschny, Oct 20 2012

Table[nn = n; a = Select[Range[nn], CoprimeQ[#, nn] &];
Mod[Apply[Times, a], nn], {n, 1, 88}] (* Geoffrey Critzer, Jan 03 2015 *)

def A160377(n):
    r = 1
    for k in (1..n):
        if gcd(n, k) == 1: r = mod(r*k, n)
    return r
[A160377(n) for n in (1..88)]  # Peter Luschny, Oct 20 2012

a(n) = A001783(n) mod n. - R. J. Mathar, May 21 2009
For n>2, a(n)=n-1 if A060594(n)=2; otherwise a(n)=1. - Max Alekseyev
a(n) = Gauss_factorial(n, n) modulo n. (Definition of the Gauss factorial in A216919.) - Peter Luschny, Oct 20 2012

Edited and extended by R. J. Mathar and Max Alekseyev, May 21 2009

A216917 Square array read by antidiagonals, T(N,n) = lcm{1<=j<=N, gcd(j,n)=1 | j} for N >= 0, n >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 12, 3, 2, 1, 1, 60, 3, 2, 1, 1, 1, 60, 15, 4, 3, 2, 1, 1, 420, 15, 20, 3, 6, 1, 1, 1, 840, 105, 20, 15, 12, 1, 2, 1, 1, 2520, 105, 140, 15, 12, 1, 6, 1, 1, 1, 2520, 315, 280, 105, 12, 5, 12, 3, 2, 1, 1, 27720, 315, 280, 105, 84

Offset: 1

Peter Luschny, Oct 02 2012

T(N,n) is the least common multiple of all integers up to N that are relatively prime to n.

Replacing LCM in the definition with "product" gives the Gauss factorial A216919.

			   n | N=0 1 2 3  4  5  6   7   8    9   10
-----+-------------------------------------
   1 |   1 1 2 6 12 60 60 420 840 2520 2520
   2 |   1 1 1 3  3 15 15 105 105  315  315
   3 |   1 1 2 2  4 20 20 140 280  280  280
   4 |   1 1 1 3  3 15 15 105 105  315  315
   5 |   1 1 2 6 12 12 12  84 168  504  504
   6 |   1 1 1 1  1  5  5  35  35   35   35
   7 |   1 1 2 6 12 60 60  60 120  360  360
   8 |   1 1 1 3  3 15 15 105 105  315  315
   9 |   1 1 2 2  4 20 20 140 280  280  280
  10 |   1 1 1 3  3  3  3  21  21   63   63
  11 |   1 1 2 6 12 60 60 420 840 2520 2520
  12 |   1 1 1 1  1  5  5  35  35   35   35
  13 |   1 1 2 6 12 60 60 420 840 2520 2520

t[, 0] = 1; t[n, k_] := LCM @@ Select[Range[k], CoprimeQ[#, n]&]; Table[t[n - k + 1, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

def A216917(N, n):
    return lcm([j for j in (1..N) if gcd(j, n) == 1])
for n in (1..13): [A216917(N,n) for N in (0..10)]

For n > 0:
A(n,1) = A003418(n);
A(n,2^k) = A217858(n) for k > 0;
A(n,3^k) = A128501(n-1) for k > 0;
A(2,n) = A000034(n);
A(3,n) = A129203(n-1);
A(4,n) = A129197(n-1);
A(n,n) = A038610(n);
A(floor(n/2),n) = A124443(n);
A(n,1)/A(n,n) = A064446(n);
A(n,1)/A(n,2) = A053644(n).

A212307 Numerator of n!/3^n.

Original entry on oeis.org

1, 1, 2, 2, 8, 40, 80, 560, 4480, 4480, 44800, 492800, 1971200, 25625600, 358758400, 1793792000, 28700672000, 487911424000, 975822848000, 18540634112000, 370812682240000, 2595688775680000, 57105153064960000, 1313418520494080000, 10507348163952640000

Offset: 0

Jean-François Alcover and Paul Curtz, Oct 30 2013

Also the 3rd column of A152656 (or of A216919).

Cf. A001316, A049606, A125824 (denominators), A152656, A216919.

Cf. A000142, A060828.

Table[Numerator[n!/3^n], {n, 0, 32}]
(* or *) CoefficientList[Series[Exp[3x], {x, 0, 32}], x] // Denominator

a(n) = numerator(n!/3^n); \\ Michel Marcus, Oct 30 2013

a(n) = Product_{i=1..n} A038502(i). - Tom Edgar, Mar 22 2014

a(n) = A000142(n)/A060828(n). - Ridouane Oudra, Sep 23 2024

A216913 a(n) = Gauss_primorial(3n, 3) / Gauss_primorial(3n, 3*n).

Original entry on oeis.org

1, 2, 1, 2, 5, 2, 7, 2, 1, 10, 11, 2, 13, 14, 5, 2, 17, 2, 19, 10, 7, 22, 23, 2, 5, 26, 1, 14, 29, 10, 31, 2, 11, 34, 35, 2, 37, 38, 13, 10, 41, 14, 43, 22, 5, 46, 47, 2, 7, 10, 17, 26, 53, 2, 55, 14, 19, 58, 59, 10, 61, 62, 7, 2, 65, 22, 67, 34, 23, 70, 71

Offset: 1

Peter Luschny, Oct 02 2012

The term Gauss primorial was introduced in A216914 and denotes the restriction of the Gauss factorial N_n! (see A216919) to prime factors.

Multiplicative because both A007947 and A109007 are. - Andrew Howroyd, Aug 02 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A007947, A065463, A109007, A204455, A216914, A216919.

[&+[EulerPhi(d)*MoebiusMu(3*d)^2:d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Oct 19 2019

Table[n/Sum[Floor[Cos[Pi k^(3 n)/(3 n)]^2], {k, 3 n}], {n, 71}] (* Michael De Vlieger, May 24 2016 *)
a[n_] := Times @@ (First /@ FactorInteger[n])/GCD[n, 3]; Array[a, 100] (* Amiram Eldar, Nov 17 2022 *)

a(n)={factorback(factor(n)[, 1])/gcd(3,n)} \\ Andrew Howroyd, Aug 02 2018

def Gauss_primorial(N, n):
    return mul(j for j in (1..N) if gcd(j, n) == 1 and is_prime(j))
def A216913(n): return Gauss_primorial(3*n, 3)/Gauss_primorial(3*n, 3*n)
[A216913(n) for n in (1..80)]

a(n) = n/Sum_{k=1..3n} floor(cos^2(Pi*k^(3n)/(3n))). - Anthony Browne, May 24 2016
a(n) = A007947(n)/A109007(n). - Andrew Howroyd, Aug 02 2018
a(n) = Sum_{d|n} phi(d)*mu(3d)^2. - Ridouane Oudra, Oct 19 2019
From Amiram Eldar, Nov 17 2022: (Start)
Multiplicative with a(3^e) = 1, and a(p^e) = p for p != 3.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (9/22) * Product_{p prime} (1 - 1/(p*(p+1))) = (9/22) * A065463 = 0.2881809... . (End)

A216918 Odd numbers with at least 3 distinct prime factors.

Original entry on oeis.org

105, 165, 195, 231, 255, 273, 285, 315, 345, 357, 385, 399, 429, 435, 455, 465, 483, 495, 525, 555, 561, 585, 595, 609, 615, 627, 645, 651, 663, 665, 693, 705, 715, 735, 741, 759, 765, 777, 795, 805, 819, 825, 855, 861, 885, 897, 903, 915, 935, 945, 957, 969

Offset: 1

Peter Luschny, Oct 02 2012

nonn

If "at least" is changed to "exactly" we get A278569. - N. J. A. Sloane, Nov 27 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000
J. B. Cosgrave, K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008), p.11.

A278569 is a subsequence.

a:= proc(n) option remember; local k;
      for k from 2+ `if`(n=1, 103, a(n-1)) by 2
        while nops(numtheory[factorset](k))<=2 do od; k
    end:
seq (a(n), n=1..100);  # Alois P. Heinz, Oct 03 2012

Select[Range[1, 999, 2], (PrimeNu[#] >= 3)&] (* Jean-François Alcover, Feb 27 2014 *)

def is_A216918(n):
    if n % 2 == 0: return False
    return len(n.prime_divisors()) >= 3
def A216918_list(n): return [k for k in srange(1, n + 1, 2) if is_A216918(k)]
A216918_list(969)

Gauss_factorial(floor(a(n)/2), a(n)) == 1 (mod a(n)). (Cf. A216919)

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

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Oct 02 2012

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.

			[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

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).

(* 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 *)

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)]

N_n# = product_{1<=j<=N, GCD(j, n) = 1, j is prime} j.

A216915 T(n, k) = Product{1<=j<=n, gcd(j,k)=1 | j} / lcm{1<=j<=n, gcd(j,k)=1 | j} for n >= 0, k >= 1, square array read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 2, 1, 1, 1, 1, 12, 1, 2, 1, 1, 1, 1, 1, 48, 1, 2, 1, 2, 1, 1, 1, 1, 144, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1440, 3, 8, 1, 12, 1, 2, 1, 1, 1, 1, 1440, 3, 8, 1, 12, 1, 2, 1, 1, 1, 1, 1, 17280, 3, 80

Offset: 1

Peter Luschny, Oct 02 2012

T(n,k) = Product(R(n,k))/lcm(R(n,k)) where R(n,k) is the set of all integers up to n that are relatively prime to k.

T(n,k) = A216919(n,k)/A216917(n,k).

			   k |n=0  1  2  3  4  5  6  7  8   9   10
  ---+------------------------------------
   1 |  1  1  1  1  2  2 12 12 48 144 1440
   2 |  1  1  1  1  1  1  1  1  1   3    3
   3 |  1  1  1  1  2  2  2  2  8   8   80
   4 |  1  1  1  1  1  1  1  1  1   3    3
   5 |  1  1  1  1  2  2 12 12 48 144  144
   6 |  1  1  1  1  1  1  1  1  1   1    1
   7 |  1  1  1  1  2  2 12 12 48 144 1440
   8 |  1  1  1  1  1  1  1  1  1   3    3
   9 |  1  1  1  1  2  2  2  2  8   8   80
  10 |  1  1  1  1  1  1  1  1  1   3    3
  11 |  1  1  1  1  2  2 12 12 48 144 1440
  12 |  1  1  1  1  1  1  1  1  1   1    1
  13 |  1  1  1  1  2  2 12 12 48 144 1440

def A216915(n, k):
    def R(n, k): return [j for j in (1..n) if gcd(j, k) == 1]
    return mul(R(n,k))/lcm(R(n, k))
for k in (1..13): [A216915(n, k) for n in (0..10)]

For n > 0:
A(n,1) = A025527(n);
A(4,n) = A000034(n);
A(n,n) = A128247(n).

cp's OEIS Frontend

A001783 n-phi-torial, or phi-torial of n: Product k, 1 <= k <= n, k relatively prime to n.

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A066570 Product of numbers <= n that have a prime factor in common with n.

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A103131 The product of the residues in [1,n] relatively prime to n, taken modulo n.

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A160377 Phi-torial of n (A001783) modulo n.

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A216917 Square array read by antidiagonals, T(N,n) = lcm{1<=j<=N, gcd(j,n)=1 | j} for N >= 0, n >= 1.

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A212307 Numerator of n!/3^n.

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A216913 a(n) = Gauss_primorial(3n, 3) / Gauss_primorial(3n, 3*n).