A002322 -id:A002322 - cp's OEIS frontend

A077197 a(n) = lambda(phi(n)) where lambda(n) (A002322) is the Carmichael lambda function which gives the smallest integer m such that k^m = 1 mod n for all integers k relatively prime to n and phi(n) is A000010.

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 6, 2, 2, 4, 10, 2, 4, 2, 6, 2, 6, 2, 4, 4, 4, 4, 2, 2, 6, 6, 2, 4, 4, 2, 6, 4, 2, 10, 22, 4, 6, 4, 8, 2, 12, 6, 4, 2, 6, 6, 28, 4, 4, 4, 6, 8, 4, 4, 10, 8, 10, 2, 12, 2, 6, 6, 4, 6, 4, 2, 12, 8, 18, 4, 40, 2, 16, 6, 6, 4, 10, 2, 6, 10, 4, 22, 6, 8

Offset: 1

Joseph L. Pe, Nov 29 2002

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000010, A002322, A206941.

Table[CarmichaelLambda[EulerPhi[n]], {n, 1, 100}]

a(n)=lcm(znstar(eulerphi(n))[2]) \\ Charles R Greathouse IV, Feb 21 2013

A173927 Smallest integer k such that the number of iterations of Carmichael lambda function (A002322) needed to reach 1 starting at k (k is counted) is n.

Original entry on oeis.org

1, 2, 3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 531441, 1594323, 4782969, 14348907, 43046721, 86093443, 258280327, 688747547

Offset: 1

Michel Lagneau, Nov 26 2010

Smallest number k such that the trajectory of k under iteration of Carmichael lambda function contains exactly n distinct numbers (including k and the fixed point).
The first 13 terms are 1 or a prime. The next five terms are powers of 3. Then another prime. What explains this behavior? - T. D. Noe, Mar 23 2012
A185816(a(n)) = n - 1. - Reinhard Zumkeller, Sep 02 2014
If a(n) (n > 1) is either a prime or a power of 3, then a(n) is also the smallest integer k such that the number of iterations of Euler's totient function (A000010) needed to reach 1 starting at k (k is counted) is n. - Jianing Song, Jul 10 2019

			for n=5, a(5)=11 gives a chain of length 5 because the trajectory is 11 -> 10 -> 4 -> 2 -> 1.

Nick Harland, The number of iterates of the Carmichael lambda function required to reach 1, arXiv:1203.4791v1 [math.NT], Mar 21 2012.

Cf. A002322, A027763, A056637.

Cf. A185816 (number of iterations of Carmichael lambda function needed to reach 1), A003434 (number of iterations of Euler's totient function needed to reach 1).

import Data.List (elemIndex); import Data.Maybe (fromJust)
a173927 = (+ 1) . fromJust . (`elemIndex` map (+ 1) a185816_list)
-- Reinhard Zumkeller, Sep 02 2014

f[n_] := Length@ NestWhileList[ CarmichaelLambda, n, Unequal, 2] - 1; t = Table[0, {30}]; k = 1; While[k < 2100000001, a = f@ k; If[ t[[a]] == 0, t[[a]] = k; Print[a, " = ", k]]; k++] (* slightly modified by Robert G. Wilson v, Sep 01 2014 *)

a(20)-a(21) from Robert G. Wilson v, Sep 01 2014

A181776 a(n) = lambda(lambda(n)), where lambda(n) is the Carmichael lambda function (A002322).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 2, 2, 2, 2, 4, 2, 6, 2, 2, 4, 10, 1, 4, 2, 6, 2, 6, 2, 4, 2, 4, 4, 2, 2, 6, 6, 2, 2, 4, 2, 6, 4, 2, 10, 22, 2, 6, 4, 4, 2, 12, 6, 4, 2, 6, 6, 28, 2, 4, 4, 2, 4, 2, 4, 10, 4, 10, 2, 12, 2, 6, 6, 4, 6, 4, 2, 12, 2, 18, 4, 40, 2, 4, 6, 6

Offset: 1

Michel Lagneau, Nov 12 2010

nonn

Harland proves the conjecture of Martin & Pomerance that a(n) = n exp ((1 + o(1))(log log n)^2 log log log n) for almost all n, as well as a generalization to k-th iterates. - Charles R Greathouse IV, Dec 21 2011

			a(11) = 4 is in the sequence because A002322(11) = 10 and A002322(10) = 4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Nick Harland, The iterated Carmichael lambda function, arXiv:1111.3667 [math.NT], 2011.
G. Martin and C. Pomerance, The iterated Carmichael lambda-function and the number of cycles of the power generator, Acta Arith. 118:4 (2005), pp. 305-335.

Cf. A002322, A002997.

Table[CarmichaelLambda[CarmichaelLambda[n]],{n,1,100}]
Table[Nest[CarmichaelLambda,n,2],{n,100}] (* Harvey P. Dale, Jul 01 2020 *)

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

A231570 Composite numbers n such that lambda(n) divides 3n-3, where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

9, 21, 45, 63, 65, 105, 117, 133, 231, 273, 341, 481, 561, 585, 645, 651, 1001, 1105, 1281, 1365, 1541, 1729, 2465, 2821, 3201, 3605, 4033, 4371, 4641, 4921, 5461, 5565, 6305, 6533, 6601, 7107, 7161, 8321, 8911, 10585, 11041, 12545, 13333, 13833, 14981

Offset: 1

José María Grau Ribas, Nov 11 2013

nonn

Conjecture: the relative asymptotic density of the Carmichael numbers in this sequence exists, is positive and smaller than 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. M. Grau and Antonio Oller-Marcén, Generalizing Giuga's conjecture, arXiv:1103.3483 [math.NT], 2011.

Cf. A002322, A231569, A231571, A231572, A231573.

Select [1 + Range[100000], ! PrimeQ[#] && IntegerQ[3 (# -1)/ CarmichaelLambda[#]] &]

is(n)=!isprime(n) && (3*n-3)%lcm(znstar(n)[2])==0 && n>1 \\ Charles R Greathouse IV, Nov 13 2013

A231572 Composite numbers n such that lambda(n) divides 5n-5, where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

25, 33, 165, 217, 325, 385, 561, 793, 825, 1025, 1045, 1065, 1105, 1353, 1525, 1705, 1729, 2465, 2665, 2821, 3565, 4123, 4681, 5005, 5185, 5425, 6601, 6697, 8029, 8569, 8911, 9073, 10585, 11005, 12025, 12505, 12801, 13237, 13741, 14707, 14725, 14905, 15457

Offset: 1

José María Grau Ribas, Nov 11 2013

nonn

Contains the Carmichael numbers (A002997).

Conjecture: the relative asymptotic density of the Carmichael numbers in this sequence exists, is positive and smaller than 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. M. Grau and Antonio Oller-Marcén, Generalizing Giuga's conjecture, arXiv:1103.3483 [math.NT], 2011.

Cf. A231569-A231575, A002322.

Select [1 + Range[100000], ! PrimeQ[#] && IntegerQ[5 (# -1)/ CarmichaelLambda[#]] &]

is(n)=!isprime(n) && (5*n-5)%lcm(znstar(n)[2])==0 && n>1 \\ Charles R Greathouse IV, Nov 13 2013

A231573 Composite numbers n such that lambda(n) divides 6n-6, where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

4, 6, 8, 9, 12, 14, 15, 18, 21, 24, 28, 35, 36, 39, 42, 45, 56, 63, 65, 66, 72, 76, 84, 91, 105, 117, 126, 133, 153, 168, 186, 195, 231, 247, 252, 259, 273, 276, 315, 341, 344, 396, 435, 455, 481, 504, 532, 561, 585, 616, 645, 651, 671, 703, 804, 819, 861

Offset: 1

José María Grau Ribas, Nov 11 2013

nonn

Contains the Carmichael numbers, A231569 and A231570.

Conjecture: the relative asymptotic density of Carmichael numbers in this sequence exists, is positive and smaller than 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. M. Grau and Antonio Oller-Marcén, Generalizing Giuga's conjecture, arXiv:1103.3483 [math.NT], 2011.

Cf. A231569-A231575, A002322.

Select [1 + Range[100000], ! PrimeQ[#] && IntegerQ[6 (# -1)/ CarmichaelLambda[#]] &]

is(n)=!isprime(n) && (6*n-6)%lcm(znstar(n)[2])==0 && n>1 \\ Charles R Greathouse IV, Nov 13 2013

A270562 a(n) is the largest number m satisfying lambda(m)=n, or zero if there is no solution, where lambda(m) is Carmichael's lambda function A002322(m).

Original entry on oeis.org

2, 24, 0, 240, 0, 504, 0, 480, 0, 264, 0, 65520, 0, 0, 0, 16320, 0, 28728, 0, 13200, 0, 552, 0, 131040, 0, 0, 0, 6960, 0, 171864, 0, 32640, 0, 0, 0, 138181680, 0, 0, 0, 1082400, 0, 151704, 0, 5520, 0, 1128, 0, 4455360, 0, 0, 0, 12720, 0, 86184, 0, 13920, 0, 1416, 0, 6814407600, 0, 0, 0, 65280

Offset: 1

Joerg Arndt, Mar 19 2016

nonn

a(n) is the largest modulus m such that the largest order of any element in the multiplicative group modulo m is n; a(n) is zero if there is no such group with largest order n.
Omitting the zeros gives A143407.
a(n) = 0 if n is not a term of A002174.

Gheorghe Coserea, Table of n, a(n) for n = 1..50005
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1910), 232-238.

Cf. A002322, A002174, A051222, A143407, A270564, A111725.

A289624 a(n) = A002322(n)/A007947(A034380(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 1, 6, 4, 10, 1, 12, 6, 2, 2, 16, 6, 18, 2, 3, 10, 22, 1, 20, 12, 18, 3, 28, 2, 30, 4, 5, 16, 6, 3, 36, 18, 6, 2, 40, 3, 42, 5, 6, 22, 46, 2, 42, 20, 8, 6, 52, 18, 10, 3, 9, 28, 58, 2, 60, 30, 1, 8, 6, 5, 66, 8, 11, 6, 70, 3, 72, 36, 10, 9, 15, 6, 78, 2, 54, 40, 82, 3, 8, 42, 14, 5, 88, 6, 2, 11, 15, 46

Offset: 1

Antti Karttunen, Jul 17 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A000010, A002322, A007947, A034380, A080400.

with(numtheory):
rad := n -> ilcm(op(factorset(n))):
a := n -> lambda(n)/rad(phi(n)/lambda(n)):
seq(a(n), n=1..94); # Peter Luschny, Jul 17 2017

a007947[n_]:=Last@ Select[Divisors[n], SquareFreeQ[#] &]; Table[Numerator[CarmichaelLambda[n]/a007947[EulerPhi[n]/CarmichaelLambda[n]]], {n, 100}] (* Indranil Ghosh, Jul 17 2017 *)

A002322(n) = lcm(znstar(n)[2]); \\ This function from Charles R Greathouse IV, Aug 04 2012
A007947(n) = factorback(factorint(n)[, 1]); \\ This function from Andrew Lelechenko, May 09 2014
A289624(n) = A002322(n)/A007947(eulerphi(n)/A002322(n));

from sage.crypto.util import carmichael_lambda
def A007947(n): return mul(p for p in prime_divisors(n))
def A000010(n): return euler_phi(n)
def A002322(n): return carmichael_lambda(n)
def A289624(n): return A002322(n)/A007947(A000010(n)/A002322(n))
print([A289624(n) for n in (1..94)]) # Peter Luschny, Jul 17 2017

a(n) = A002322(n) / A007947(A034380(n)) = A002322(n) / A007947(A000010(n) / A002322(n)).

A306414 a(n) = psi(A002997(n)), psi = A002322.

Original entry on oeis.org

80, 48, 36, 112, 60, 1320, 198, 504, 360, 180, 120, 288, 1224, 2024, 36, 240, 300, 720, 72, 2320, 60, 108, 600, 112, 216, 1980, 8568, 528, 420, 6120, 2268, 360, 540, 3360, 240, 4900, 2160, 396, 432, 144, 300, 120, 576, 2970, 612, 396, 648, 630, 1260, 1080, 1008

Offset: 1

Jianing Song, Feb 13 2019

nonn

By definition, A002997 lists the composite numbers k such that psi(k) divides k - 1. This sequence lists the values for psi(k) when k runs through A002997.

			A002997(1) = 561, psi(561) = 80, 561 = 7*80 + 1.
A002997(2) = 1105, psi(1105) = 48, 1105 = 23*48 + 1.
A002997(3) = 1729, psi(1729) = 36, 1729 = 48*36 + 1.
A002997(4) = 2465, psi(2465) = 112, 2465 = 22*112 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002997, A002322, A174590.

seq[kmax_] := Module[{s = {}, lam}, Do[If[CompositeQ[k], lam = CarmichaelLambda[k]; If[Mod[k, lam] == 1, AppendTo[s, lam]]], {k, 9, kmax, 2}]; s]; seq[10^6] (* Amiram Eldar, Nov 14 2023 *)

forstep(n=3, 1e6, 2, my(m=lcm(znstar(n)[2])); if((n-1)%m==0 && !isprime(n), print1(m, ", ")))

a(n) = A002322(A002997(n)).

a(n) = (A002997(n) - 1) / A174590(n). - Jianing Song, Dec 12 2021

A317210 Composite numbers k+1 such that A002322(A027760(k)) = k.

Original entry on oeis.org

21, 45, 49, 81, 85, 91, 93, 111, 117, 121, 133, 141, 145, 165, 175, 177, 201, 205, 209, 213, 217, 221, 231, 235, 247, 253, 261, 265, 273, 289, 291, 301, 309, 319, 325, 333, 357, 361, 365, 369, 381, 391, 411, 415, 441, 445, 451, 453, 465, 469, 477, 481, 493

Offset: 1

Max Alekseyev and Thomas Ordowski, Jul 09 2018

nonn

Also, composite numbers n such that LCM( p-1 : prime p|A027642(n-1) ) = n-1. Also, composite numbers n such that LCM( p-1 : p is prime & (p-1)|(n-1) ) = n-1. - Max Alekseyev, Dec 03 2021

Contains all Carmichael numbers except 2628073, 3224065, 23382529, 182356993, 1419339691, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A027760, A002322.

1 + Select[Range[500], CompositeQ[# + 1] && CarmichaelLambda[ Times @@ Select[1 + Divisors@ #, PrimeQ]] == # &] (* Giovanni Resta, Aug 13 2018 *)

a027760(n) = denominator(sumdiv(n, d, if(isprime(d+1), 1/(d+1))));
a002322(n) = lcm(znstar(n)[2]);
isok(n) = !isprime(n) && (n--) && !frac(a002322(a027760(n))/n); \\ Michel Marcus, Aug 13 2018

More terms from Giovanni Resta, Aug 13 2018

cp's OEIS Frontend

A077197 a(n) = lambda(phi(n)) where lambda(n) (A002322) is the Carmichael lambda function which gives the smallest integer m such that k^m = 1 mod n for all integers k relatively prime to n and phi(n) is A000010.

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A173927 Smallest integer k such that the number of iterations of Carmichael lambda function (A002322) needed to reach 1 starting at k (k is counted) is n.

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A181776 a(n) = lambda(lambda(n)), where lambda(n) is the Carmichael lambda function (A002322).

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A231570 Composite numbers n such that lambda(n) divides 3n-3, where lambda is the Carmichael lambda function (A002322).

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A231572 Composite numbers n such that lambda(n) divides 5n-5, where lambda is the Carmichael lambda function (A002322).

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A231573 Composite numbers n such that lambda(n) divides 6n-6, where lambda is the Carmichael lambda function (A002322).

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A270562 a(n) is the largest number m satisfying lambda(m)=n, or zero if there is no solution, where lambda(m) is Carmichael's lambda function A002322(m).

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A289624 a(n) = A002322(n)/A007947(A034380(n)).

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