A124441 -id:A124441 - 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

A216919 The Gauss factorial N_n! 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, 24, 3, 2, 1, 1, 120, 3, 2, 1, 1, 1, 720, 15, 8, 3, 2, 1, 1, 5040, 15, 40, 3, 6, 1, 1, 1, 40320, 105, 40, 15, 24, 1, 2, 1, 1, 362880, 105, 280, 15, 24, 1, 6, 1, 1, 1, 3628800, 945, 2240, 105, 144, 5, 24, 3, 2, 1, 1, 39916800, 945

Offset: 1

Peter Luschny, Oct 01 2012

The term is due to Cosgrave & Dilcher. The Gauss factorial should not be confused with the q-factorial [n]_q! which is also called Gaussian factorial.

			[n\N][0, 1, 2, 3,  4,   5,   6,    7,     8,      9,     10]
------------------------------------------------------------
[  1] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 [A000142]
[  2] 1, 1, 1, 3,  3,  15,  15,  105,   105,    945,     945 [A055634, A133221]
[  3] 1, 1, 2, 2,  8,  40,  40,  280,  2240,   2240,   22400 [A232980]
[  4] 1, 1, 1, 3,  3,  15,  15,  105,   105,    945,     945
[  5] 1, 1, 2, 6, 24,  24, 144, 1008,  8064,  72576,   72576 [A232981]
[  6] 1, 1, 1, 1,  1,   5,   5,   35,    35,     35,      35 [A232982]
[  7] 1, 1, 2, 6, 24, 120, 720,  720,  5760,  51840,  518400 [A232983]
[  8] 1, 1, 1, 3,  3,  15,  15,  105,   105,    945,     945
[  9] 1, 1, 2, 2,  8,  40,  40,  280,  2240,   2240,   22400
[ 10] 1, 1, 1, 3,  3,   3,   3,   21,    21,    189,     189 [A232984]
[ 11] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 [A232985]
[ 12] 1, 1, 1, 1,  1,   5,   5,   35,    35,     35,      35
[ 13] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800

Alois P. Heinz, Antidiagonals n = 1..141, flattened
J. B. Cosgrave, K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008).
J. B. Cosgrave, K. Dilcher, An Introduction to Gauss Factorials, The American Mathematical Monthly, Vol. 118, No. 9 (2011), 812-829.
K. Dilcher, Gauss Factorials: Properties and Applications. Video by the Irmacs Centre, May 18, 2011.

A000142(n) = n! = Gauss_factorial(n, 1).
A001147(n) = Gauss_factorial(2*n, 2).
A055634(n) = Gauss_factorial(n, 2)*(-1)^n.
A001783(n) = Gauss_factorial(n, n).
A124441(n) = Gauss_factorial(floor(n/2), n).
A124442(n) = Gauss_factorial(n, n)/Gauss_factorial(floor(n/2), n).
A066570(n) = Gauss_factorial(n, 1)/Gauss_factorial(n, n).
Cf. A133221, A232980, A232981, A232982, A232983, A232984, A232985.

A:= (n, N)-> mul(`if`(igcd(j, n)=1, j, 1), j=1..N):
seq(seq(A(n, d-n), n=1..d), d=1..12);  # Alois P. Heinz, Oct 03 2012

GaussFactorial[m_, n_] := Product[ If[ GCD[j, n] == 1, j, 1], {j, 1, m}]; Table[ GaussFactorial[m - n, n], {m, 1, 12}, {n, 1, m}] // Flatten (* Jean-François Alcover, Mar 18 2013 *)

T(m,n)=prod(k=2, m, if(gcd(k,n)==1, k, 1))
for(s=1,10,for(n=1,s,print1(T(s-n,n)", "))) \\ Charles R Greathouse IV, Oct 01 2012

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

N_n! = product_{1<=j<=N, GCD(j,n)=1} j.

A124442 a(n) = Product_{ceiling(n/2) <= k <= n, gcd(k,n)=1} k.

Original entry on oeis.org

1, 1, 2, 3, 12, 5, 120, 35, 280, 63, 30240, 77, 665280, 1287, 16016, 19305, 518918400, 2431, 17643225600, 46189, 14780480, 1322685, 28158588057600, 96577, 4317650168832, 58503375, 475931456000, 75218625, 3497296636753920000, 215441, 202843204931727360000

Offset: 1

Leroy Quet, Nov 01 2006

nonn

			The integers which are >= 9/2 and are <= 9 and which are coprime to 9 are 5, 7 and 8. So a(9) = 5*7*8 = 280.

Alois P. Heinz, Table of n, a(n) for n = 1..300
J. B. Cosgrave, K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8(2008)

Cf. A124441.

a:=proc(n) local b,k: b:=1: for k from ceil(n/2) to n do if gcd(k,n)=1 then b:=b*k else b:=b fi od: b; end: seq(a(n),n=1..33); # Emeric Deutsch, Nov 03 2006

f[n_] := Times @@ Select[Range[Ceiling[n/2], n], GCD[ #, n] == 1 &];Table[f[n], {n, 30}] (* Ray Chandler, Nov 12 2006 *)

A124442(n)=prod(k=(n+1)\2,n-1, k^(gcd(k, n)==1))  \\ M. F. Hasler, Jul 23 2011

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A124442(n): return Gauss_factorial(n, n)/Gauss_factorial(n//2, n)
[A124442(n) for n in (1..29)] # Peter Luschny, Oct 01 2012

a(n) = A001783(n)/A124441(n). - M. F. Hasler, Jul 23 2011

More terms from Emeric Deutsch, Nov 03 2006

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

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A216919 The Gauss factorial N_n! for N >= 0, n >= 1, square array read by antidiagonals.

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A124442 a(n) = Product_{ceiling(n/2) <= k <= n, gcd(k,n)=1} k.

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A193340 Terms of A001783 which are smaller than the preceding term in that sequence.

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