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A001783 n-phi-torial, or phi-torial of n: Product k, 1<=k<=n, k relatively prime to n.
(Formerly M0921 N0346)
25
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 (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

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.

Problem E1045, Amer. Math. Monthly, 60 (1953), 422.

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).

L. Toth, Weighted Gcd-Sum Functions, Journal of Integer Sequences, 14 (2011), #11.7.7.

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, K. Dilcher, The multiplicative orders of certain Gauss factorials, Intl. J. Number Theory 7 (1) (2011) 145-171.

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

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

(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.

Sequence in context: A115031 A030418 A037277 * A095996 A061098 A160630

Adjacent sequences:  A001780 A001781 A001782 * A001784 A001785 A001786

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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Last modified April 18 19:08 EDT 2014. Contains 240733 sequences.