A141258 -id:A141258 - 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

A061258 a(n) = Sum_{d|n} d*psi(d), where psi(d) is reduced totient function, cf. A002322.

Original entry on oeis.org

1, 3, 7, 11, 21, 21, 43, 27, 61, 63, 111, 53, 157, 129, 87, 91, 273, 183, 343, 151, 175, 333, 507, 117, 521, 471, 547, 305, 813, 261, 931, 347, 447, 819, 483, 431, 1333, 1029, 631, 327, 1641, 525, 1807, 781, 681, 1521, 2163, 373, 2101, 1563, 1095, 1103, 2757

Offset: 1

Vladeta Jovovic, Apr 21 2001

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002322, A057660, A001001, A060640, A061259.

Cf. A000005, A027750, A141258.

a061258 n = sum $ zipWith (*) ds $ map a002322 ds
            where ds = a027750_row n
-- Reinhard Zumkeller, Sep 02 2014

a[n_] := DivisorSum[n, # * CarmichaelLambda[#] &]; Array[a, 100] (* Amiram Eldar, Apr 13 2024 *)

a(n) = sumdiv(n, d, d * lcm(znstar(d)[2])); \\ Amiram Eldar, Apr 13 2024

a(n) = Sum_{k = 1..A000005(n)} (A027750(n,k)*A002322(A027750(n,k))). - Reinhard Zumkeller, Sep 02 2014

A131492 Numbers n such that the sum of the Carmichael lambda functions of the divisors is a proper divisor of n.

Original entry on oeis.org

140, 189, 378, 1375, 2750, 2775, 2997, 4524, 5550, 5661, 5994, 6375, 11253, 11322, 12750, 13416, 13505, 22506, 25925, 27010, 27511, 30613, 32208, 32513, 32760, 45917, 49665, 49959, 51850, 55022, 61061, 61226, 65026, 67488, 91834, 93605

Offset: 1

R. J. Mathar, Jul 29 2007

nonn

The auxiliary sequence defined by b(n)=sum_{d|n} A002322(d) starts 1,2,3,4,5,6,7,6,9,10,11,10,13,14,11,10,17,18,19,16,...

The auxiliary sequence is A141258. [Reinhard Zumkeller, Feb 17 2012]

Amiram Eldar, Table of n, a(n) for n = 1..1000
W. D. Banks and F. Luca, On integers with a special divisibility property, Archivum Mathematicum (BRNO) 42 (2006) pp 31-42.

Cf. A002322, A039716.

Select[ Range[100000], Divisible[#, s = Total[ CarmichaelLambda /@ Divisors[#]]] && s < # &] (* Jean-François Alcover, Jun 24 2013 *)

lambda(p,alpha)={ if(p>=3 || alpha<=2, return(p^(alpha-1)*(p-1)), return(2^(alpha-2)) ; ) ; } A002322(n)={ local(pf,rmax,resul) ; if(n==1, return(1) ) ; pf=factor(n) ; rmax=matsize(pf)[1] ; resul= lambda(pf[1,1],pf[1,2]) ; for(r=2,rmax, resul=lcm(resul,lambda(pf[r,1],pf[r,2])) ; ) ; return(resul) ; } b(n)={ sumdiv(n,d,A002322(d)) ; } { for(n=1,120000, l=b(n) ; if( l != 1 && l != n && n%l==0, print1(n,",") ) ; ) ; }

n such that (sum_{d|n} A002322(d)) | n.

A239641 Dirichlet inverse of Carmichael lambda function (A002322).

Original entry on oeis.org

1, -1, -2, -1, -4, 2, -6, 1, -2, 4, -10, 4, -12, 6, 12, -1, -16, 2, -18, 8, 18, 10, -22, -4, -4, 12, -2, 12, -28, -12, -30, -3, 30, 16, 36, 0, -36, 18, 36, -8, -40, -18, -42, 20, 4, 22, -46, 0, -6, 4, 48, 24, -52, 2, 60, -12, 54, 28, -58, -40, -60, 30, 18, -1, 84, -30, -66, 32, 66, -36, -70, 0

Offset: 1

José María Grau Ribas, Mar 23 2014

sign

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A002322, A141258.

inverse[F_][1] := 1/F[1]; inverse[F_][n_] := -1/F[1] Sum[F[n/Divisors[n][[i]]]*inverse[F][Divisors[n][[i]]], {i, Length[Divisors[n]] - 1}];Table[inverse[CarmichaelLambda][n], {n, 1, 331}];

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, lcm(znstar(n).cyc)))} \\ Andrew Howroyd, Aug 05 2018

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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A061258 a(n) = Sum_{d|n} d*psi(d), where psi(d) is reduced totient function, cf. A002322.

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A131492 Numbers n such that the sum of the Carmichael lambda functions of the divisors is a proper divisor of n.

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A239641 Dirichlet inverse of Carmichael lambda function (A002322).

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Mathematica

PARI