A061258 -id:A061258 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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

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

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

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

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

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

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

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

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

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

A061257 Euler transform of reduced totient function psi(n), cf. A002322.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 21, 37, 58, 96, 153, 243, 376, 584, 897, 1353, 2046, 3060, 4552, 6714, 9862, 14386, 20898, 30198, 43427, 62159, 88600, 125804, 177881, 250615, 351819, 492203, 686294, 953954, 1321902, 1826394, 2516364, 3457332, 4737576, 6475332

Offset: 0

Vladeta Jovovic, Apr 21 2001

T. D. Noe, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A002322, A061258, A006171, A001970, A061255, A061256.

nn = 20; b = Table[CarmichaelLambda[n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)

G.f.: Product_{k=1..infinity} (1 - x^k)^(-psi(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*psi(d), cf. A061258.

A141258 Inverse Mobius transform of the Carmichael lambda function.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 6, 9, 10, 11, 10, 13, 14, 11, 10, 17, 18, 19, 16, 15, 22, 23, 14, 25, 26, 27, 22, 29, 22, 31, 18, 23, 34, 23, 28, 37, 38, 27, 22, 41, 30, 43, 34, 29, 46, 47, 22, 49, 50, 35, 40, 53, 54, 35, 30, 39, 58, 59, 34, 61, 62, 27, 34, 29, 46, 67, 52, 47, 46, 71, 38

Offset: 1

Gary W. Adamson, Jun 18 2008

nonn

n-th term = prime when n is prime.
This sequence is used in A131492 as an auxiliary sequence. - Reinhard Zumkeller, Feb 17 2012
a(n) = Sum_{k = 1..A000005(n)} A002322(A027750(n,k)). - Reinhard Zumkeller, Sep 02 2014

			a(6) = 6 = (1, 1, 1, 0, 0, 1) dot (1, 1, 2, 2, 4, 2) = (1 + 1 + 2 + 0 + 0 + 2); where (1, 1, 1, 0, 0, 1) = row 6 of triangle A051731.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
W. D. Banks and F. Luca, On integers with a special divisibility property, Archivum Mathematicum (BRNO) 42 (2006) pp 31-42.

Cf. A002322, A027750, A051731, A061258, A131492.

a141258 = sum . map a002322 . a027750_row
-- Reinhard Zumkeller, Sep 02 2014

A141258[n_] := DivisorSum[n, CarmichaelLambda[#]&];
Table[A141258[n],{n,1,20}] (* Enrique Pérez Herrero, Apr 22 2014 *)

a(n) = sumdiv(n, d, lcm(znstar(d)[2])); \\ see PARI script in A002322; Michel Marcus, Apr 22 2014

a(n) = Sum_{d|n} A002322(d).

More terms from R. J. Mathar, Jan 19 2009

cp's OEIS Frontend

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.

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A061257 Euler transform of reduced totient function psi(n), cf. A002322.

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A141258 Inverse Mobius transform of the Carmichael lambda function.

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A061259 a(n)=Sum_{d|n} d*numbpart(d), where numbpart(d)=number of partitions of d, cf. A000041.

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