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Search: a124441 -id:a124441
Displaying 1-4 of 4 results found.
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A216919
The Gauss factorial N_n! for N >= 0, n >= 1, square array read by antidiagonals.
+0
11
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
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
LINKS
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.
FORMULA
N_n! = product_{1<=j<=N, GCD(j,n)=1} j.
EXAMPLE
[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
MAPLE
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
MATHEMATICA
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 *)
PROG
(Sage)
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)]
(PARI) 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
CROSSREFS
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.
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Oct 01 2012
STATUS
approved
A193340
Terms of A001783 which are smaller than the preceding term in that sequence.
+0
0
1, 5, 105, 189, 385, 19305, 85085, 8729721, 1249937325, 37182145, 608142583125, 1452095555625, 215656441, 191898783962510625, 372509404162520625, 29248404810625, 431620764875678503125, 4873615036539089841, 181101347337625, 1553338924739899476440625
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Appears to be a subsequence of A124441.
LINKS
Table of n, a(n) for n=1..20.
PROG
(PARI) m=9; for(n=1, 99, m+0>(m=A001783(n)) && print1(m", "))
CROSSREFS
Cf. A193338, A193339 for record values in A001783.
KEYWORD
nonn
AUTHOR
M. F. Hasler, Jul 23 2011
STATUS
approved
A124442
a(n) = Product_{ceiling(n/2) <= k <= n, gcd(k,n)=1} k.
+0
4
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
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)
FORMULA
a(n) = A001783(n)/A124441(n). - M. F. Hasler, Jul 23 2011
EXAMPLE
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.
MAPLE
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
MATHEMATICA
f[n_] := Times @@ Select[Range[Ceiling[n/2], n], GCD[ #, n] == 1 &]; Table[f[n], {n, 30}] (* Ray Chandler, Nov 12 2006 *)
PROG
(PARI) A124442(n)=prod(k=(n+1)\2, n-1, k^(gcd(k, n)==1)) \\ M. F. Hasler, Jul 23 2011
(Sage)
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
CROSSREFS
Cf. A124441.
KEYWORD
nonn
AUTHOR
Leroy Quet, Nov 01 2006
EXTENSIONS
More terms from Emeric Deutsch, Nov 03 2006
STATUS
approved
A001783
n-phi-torial, or phi-torial of n: Product k, 1 <= k <= n, k relatively prime to n.
(Formerly M0921 N0346)
+0
36
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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
REFERENCES
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).
LINKS
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
FORMULA
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
MAPLE
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
MATHEMATICA
A001783[n_]:=Times@@Select[Range[n], CoprimeQ[n, #]&];
Array[A001783, 20] (* Enrique Pérez Herrero, Jul 23 2011 *)
PROG
(PARI) A001783(n)=prod(k=2, n-1, k^(gcd(k, n)==1)) \\ M. F. Hasler, Jul 23 2011
(PARI) 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
(Haskell)
a001783 = product . a038566_row
-- Reinhard Zumkeller, Mar 04 2012, Aug 26 2011
(Sage)
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
CROSSREFS
Cf. A023896, A066570, A038566, A124441.
Main diagonal gives A216919.
KEYWORD
nonn,nice,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from James A. Sellers, Dec 23 1999
STATUS
approved
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