A002322 Reduced totient function psi(n): least k such that x^k == 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of unit group mod n); also called the universal exponent of n.

1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 4, 4, 16, 6, 18, 4, 6, 10, 22, 2, 20, 12, 18, 6, 28, 4, 30, 8, 10, 16, 12, 6, 36, 18, 12, 4, 40, 6, 42, 10, 12, 22, 46, 4, 42, 20, 16, 12, 52, 18, 20, 6, 18, 28, 58, 4, 60, 30, 6, 16, 12, 10, 66, 16, 22, 12, 70, 6, 72, 36, 20, 18, 30, 12, 78, 4, 54

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• core
• easy
• nice

Comments

a(n) is the largest order of any element in the multiplicative group modulo n. - Joerg Arndt, Mar 19 2016

Largest period of repeating digits of 1/n written in different bases (i.e., largest value in each row of square array A066799 and least common multiple of each row). - Henry Bottomley, Dec 20 2001

References

• D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
• W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 53.
• Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 269.
• 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..10000
• L. Blum, M. Blum, and M. Shub, A simple unpredictable pseudorandom number generator, SIAM J. Comput. 15 (1986), no. 2, 364-383. see p. 377.
• P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots
• R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.
• A. Cauchy, Mémoire sur la résolution des équations indéterminées du premier degré en nombres entiers, Oeuvres Complètes. Gauthier-Villars, Paris, 1882-1938, Series (2), Vol. 12, pp. 9-47.
• A. de Vries, The prime factors of an integer(along with Euler's phi and Carmichael's lambda functions), Applet
• Paul Erdős, Carl Pomerance, and Eric Schmutz, Carmichael's lambda function, Acta Arithmetica 58 (1991), pp. 363-385.
• J.-H. Evertse and E. van Heyst, Which new RSA signatures can be computed from some given RSA signatures?, Proceedings of Eurocrypt '90, Lect. Notes Comput. Sci., 473, Springer-Verlag, pp. 84-97, see page 86.
• J. M. Grau and A. M. Oller-Marcén, On the congruence sum_{j=1}^{n-1} j^{k(n-1)} == -1 (mod n); k-strong Giuga and k-Carmichael numbers, arXiv preprint arXiv:1311.3522 [math.NT], 2013.
• Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.
• P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 80.
• Benjamin Schreyer, Rigged Horse Numbers and their Modular Periodicity, arXiv:2409.03799 [math.CO], 2024. See p. 12.
• Eric Weisstein's World of Mathematics, Carmichael Function
• Wikipedia, Carmichael function
• Wolfram Research, First 50 values of Carmichael lambda(n)
• Index entries for "core" sequences

Crossrefs

Cf. A011773, A002174, A002616, A034380, A061258, A062373, A141258, A162578 (partial sums), A218342.

Programs

Haskell

a002322 n = foldl lcm 1 $ map (a207193 . a095874) $
                          zipWith (^) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Feb 16 2012

Magma

[1] cat [ CarmichaelLambda(n) : n in [2..100]];

Maple

with(numtheory); A002322 := lambda; [seq(lambda(n), n=1..100)];

Mathematica

Table[CarmichaelLambda[k], {k, 50}] (* Artur Jasinski, Apr 05 2008 *)

PARI

A002322(n)= lcm( apply( f -> (f[1]-1)*f[1]^(f[2]-1-(f[1]==2 && f[2]>2)), Vec(factor(n)~))) \\ M. F. Hasler, Jul 05 2009

PARI

a(n)=lcm(znstar(n)[2]) \\ Charles R Greathouse IV, Aug 04 2012

Python

from sympy import reduced_totient
def A002322(n): return reduced_totient(n) # Chai Wah Wu, Feb 24 2021

Formula

If M = 2^e*P1^e1*P2^e2*...*Pk^ek, lambda(2^e) = 2^(e-1) if e=1 or 2, = 2^(e-2) if e > 2; lambda(M) = lcm(lambda(2^e), (P1-1)*P1^(e1-1), (P2-1)*P2^(e2-1), ..., (Pk-1)*Pk^(ek-1)).

a(n) = lcm_{k=1..A001221(n)} A207193(A095874(A027748(n,k)^A124010(n,k))). - Reinhard Zumkeller, Feb 16 2012