A162578 -id:A162578 - 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

A218342 Decimal expansion of e^-gamma * Product_(1 - 1/(p^3 - p^2 - p + 1)) where the product is over all primes p.

Original entry on oeis.org

3, 4, 5, 3, 7, 2, 0, 6, 4, 1, 0, 2, 9, 8, 6, 4, 8, 7, 6, 7, 3, 4, 9, 6, 8, 2, 7, 8, 9, 1, 0, 3, 3, 7, 1, 0, 7, 2, 0, 6, 6, 5, 6, 2, 5, 3, 8, 0, 4, 1, 5, 8, 7, 2, 0, 5, 6, 0, 0, 4, 8, 9, 6, 6, 2, 5, 2, 6, 5, 3, 1, 9, 5, 0, 2, 2, 5, 1, 8, 6, 6, 9, 4, 7, 9, 0, 9, 1, 1, 6, 1, 3, 9, 2, 2, 7, 6, 3, 9, 6, 9, 6, 4, 4, 7

Offset: 0

Charles R Greathouse IV, Oct 26 2012

The average order of Carmichael's lambda function is x/log x * exp(B log log x/log log log x (1 + o(1))), where B is this constant. Under the GRH, the same applies to A036391(n)/n, the sum of the orders mod n of the numbers coprime to n divided by n.

			0.34537206410298648767349682789103371072066562538041...

Paul Erdős, Carl Pomerance, and Eric Schmutz, Carmichael's lambda function, Acta Arithmetica 58 (1991), pp. 363-385.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C9).
Sungjin Kim, On the order of 'a' modulo 'n' on average, International Journal of Number Theory, Vol. 12, No. 8 (2016), pp. 2073-2080; arXiv preprint, arXiv:1509.03768 [math.NT], 2015-2016.
Pär Kurlberg and Carl Pomerance, On a problem of Arnold: the average multiplicative order of a given integer, Algebra & Number Theory, Vol. 7, No. 4 (2013), pp. 981-999; arXiv preprint, arXiv:1108.5209 [math.NT], 2012.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011; Eq. (106) page 17.

Cf. A002322, A036391, A080130, A162578.

$MaxExtraPrecision = 200; m0 = 1000; dm = 200; digits = 105; Clear[f]; f[m_] := f[m] = (slog = Normal[Series[Log[1 - 1/((p - 1)^2*(p + 1))], {p, Infinity, m}]]; Exp[slog] /. Power[p, n_] -> PrimeZetaP[-n] // N[#, digits + 10] &); f[m = m0]; Print[m, " ", f[m]]; f[m = m + dm]; While[Print[m, " ", f[m]]; RealDigits[f[m], 10, digits + 5] !=  RealDigits[f[m - dm], 10, digits + 5], m = m + dm]; B = Exp[-EulerGamma]*f[m]; RealDigits[B, 10, digits] // First (* Jean-François Alcover, Sep 20 2015 *)

exp(-Euler) * prodeulerrat(1-1/((p-1)^2*(p+1))) \\ Amiram Eldar, Mar 09 2021

More digits from Jean-François Alcover, Sep 20 2015

A242925 Numbers k such that lambda(k) divides Sum_{j=1..k} lambda(j).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 11, 12, 14, 15, 16, 18, 19, 20, 24, 30, 31, 34, 40, 42, 44, 60, 72, 80, 83, 130, 132, 136, 195, 208, 218, 232, 254, 258, 259, 260, 264, 272, 276, 305, 306, 408, 420, 440, 464, 504, 560, 585, 586, 594, 595, 609, 624, 636, 715, 819, 840

Offset: 1

Michel Lagneau, May 26 2014

nonn

Numbers k such that A162578(k)/A002322(k) = Sum_{j=1..k}A002322(j)/ A002322(k) is an integer where lambda(k) is the Carmichael lambda function (A002322).

The corresponding integers are 1, 2, 2, 3, 6, 3, 10, 4, 21, 10, 16, 17, 15, 6, 28, 76, 60, 9, 19, 98, ...

			12 is in the sequence because A162578(12)/A002322(12) = 42/2 = 21 is an integer.

G. C. Greubel, Table of n, a(n) for n = 1..1500

Cf. A194855, A002322, A162578.

with(numtheory):nn:=2000:for n from 1 to nn do:p:=lambda(n): s:=sum('lambda(j)', 'j'=1..n):if irem(s,p)=0 then printf(`%d, `,n):else fi:od:

nn = 2000; sums = Accumulate[CarmichaelLambda[Range[nn]]]; Select[Range[nn], Mod[sums[[#]],CarmichaelLambda[#]] == 0 &]

A227975 Numbers m such that m divides Sum_{k=1..m} lambda(k).

Original entry on oeis.org

1, 2, 5, 6, 10, 18, 30, 82, 4866, 8784, 10170, 23364, 76296, 247166, 585570, 735480, 848754, 1559520, 2884840, 11272940, 35642420, 56652788, 174935486, 196398413, 679063441, 1398826844, 1542228164, 1665703953, 2699813692, 5734751503

Offset: 1

Michel Lagneau, Jun 17 2016

nonn

lambda(n) is the Carmichael lambda function (A002322). The corresponding ratios (Sum_{k=1..m} lambda(k))/m are given by the sequence {1, 1, 2, 2, 3, 5, 8, 19, 711, 1221, 1399, 3011, 9034, 27187, 61246, 75971, 86971, 154710, 277344, 1015576,...}.

a(31) > 10^10. - Dana Jacobsen, Jul 07 2016

			5 is in the sequence because 5 divides Sum_{k=1..5} lambda(k) = 1 + 1 + 2 + 2 + 4 = 2*5.

Cf. A002322, A048290, A162578.

s = 0; Do[s = s + CarmichaelLambda[n]; If[IntegerQ[s/n], Print[n]], {n, 1, 10^9}]

use ntheory ":all"; my $v=0; for my $m (1..1e6) { $v=vecsum($v,carmichael_lambda($m)); say $m unless $v % $m; } # Dana Jacobsen, Jul 07 2016

More terms from Dana Jacobsen, Jul 07 2016

A306762 Smallest integer k such that Sum_(i=1..k) lambda(i) is divisible by n, where lambda(i) is the Carmichael lambda function.

Original entry on oeis.org

1, 2, 4, 3, 5, 4, 12, 11, 7, 5, 49, 6, 9, 12, 10, 15, 16, 7, 24, 8, 12, 49, 26, 30, 23, 9, 13, 17, 55, 10, 58, 15, 71, 16, 44, 19, 169, 24, 100, 11, 48, 12, 25, 49, 18, 26, 38, 30, 40, 23, 164, 28, 50, 13, 141, 20, 47, 55, 21, 14, 80, 58, 192, 15, 110, 71, 76

Offset: 1

Michel Lagneau, Mar 08 2019

			a(7) = 12 because Sum_{i=1..12} lambda(i) = 1 + 1 + 2 + 2 + 4 + 2 + 6 + 2 + 6 + 4 + 10 + 2 = 42, and 42/7 = 6.

Cf. A002322 (Carmichael lambda), A162578 (partial sums of A002322).

Cf. A053049 (analog with totient function).

S:= ListTools:-PartialSums(map(numtheory:-lambda, [$1..500])):
N:= 100: count:= 0: V:= Vector(N):
for n from 1 to 500 while count < N do
   d:= select(t -> t <= N and V[t] = 0, numtheory:-divisors(S[n]));
   count:= count + nops(d);
   V[convert(d,list)]:= n;
od:
convert(V,list); # Robert Israel, Mar 11 2019

a[n_] := (m = 1; While[! IntegerQ[Sum[CarmichaelLambda[k], {k, 1, m}]/n], m++]; m); a /@ Range[80]

lambda(n) = lcm(znstar(n)[2]);
a(n) = {my(k=1, s=lambda(k)); while (s % n, k++; s += lambda(k)); k;} \\ Michel Marcus, Mar 09 2019

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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A218342 Decimal expansion of e^-gamma * Product_(1 - 1/(p^3 - p^2 - p + 1)) where the product is over all primes p.

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A242925 Numbers k such that lambda(k) divides Sum_{j=1..k} lambda(j).

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A227975 Numbers m such that m divides Sum_{k=1..m} lambda(k).

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A306762 Smallest integer k such that Sum_(i=1..k) lambda(i) is divisible by n, where lambda(i) is the Carmichael lambda function.

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