is the smallest integer such that for all Relatively Prime to . It can be defined recursively by

Some special values are

for , and

for an Odd Prime and . The Order of (mod ) is at most (Ribenboim 1989). The values of for the first few are 1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, ... (Sloane's A002322).

**References**

Ribenboim, P. *The Book of Prime Number Records, 2nd ed.* New York: Springer-Verlag, p. 27, 1989.

Riesel, H. ``Carmichael's Function.'' *Prime Numbers and Computer Methods for Factorization, 2nd ed.*
Boston, MA: Birkhäuser, pp. 273-275, 1994.

Sloane, N. J. A. Sequence
A002322/M0298
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Vardi, I. *Computational Recreations in Mathematica.* Redwood City, CA: Addison-Wesley, p. 226, 1991.

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1999-05-26