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A002322
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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.
(Formerly M0298 N0110)
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105
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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
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OFFSET
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1,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
L. Blum; M. Blum; M. Shub, A simple unpredictable pseudorandom number generator, SIAM J. Comput. 15 (1986), no. 2, 364-383. see p. 377.
A. Cauchy, Mémoire sur la résolution des équations indéterminees 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
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, 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.
Eric Weisstein's World of Mathematics, Carmichael Function
Wikipedia, Carmichael function
Wolfram Research, First 50 vaues of Carmichael lambda(n)
Index entries for "core" sequences
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FORMULA
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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(p) = p-1 for prime p. - Paolo P. Lava, Oct 02 2006
a(n) = LCM(A207193(A095874(A027748(n,k)^A124010(n,k))): k=1..A001221(n)). - Reinhard Zumkeller, Feb 16 2012
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MAPLE
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with(numtheory); A002322 := lambda; [seq(lambda(n), n=1..100)];
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MATHEMATICA
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Table[CarmichaelLambda[k], {k, 50}] (* Artur Jasinski, Apr 05 2008 *)
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PROG
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(MAGMA) [1] cat [ CarmichaelLambda(n) : n in [2..100]];
(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
(Haskell)
a002322 n = foldl lcm 1 $ map (a207193 . a095874) $
zipWith (^) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Feb 16 2012
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CROSSREFS
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Cf. A011773, A002174, A002616, A034380, A061258, A062373, A002616.
Cf. A141258, A034380.
Sequence in context: A122457 A139770 A140635 * A127835 A117004 A128982
Adjacent sequences: A002319 A002320 A002321 * A002323 A002324 A002325
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KEYWORD
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nonn,core,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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