|
| |
|
|
A001783
|
|
n-phi-torial, or phi-torial of n: Product k, 1<=k<=n, k relatively prime to n.
(Formerly M0921 N0346)
|
|
21
| |
|
|
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; 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.
|
|
|
REFERENCES
| 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, 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
|
|
|
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; [From 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 n = product [totative | totative <- [1..n], gcd n totative == 1]
-- Reinhard Zumkeller, Aug 26 2011
|
|
|
CROSSREFS
| Cf. A023896, A066570.
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 (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 23 1999
|
| |
|
|
