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.

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

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..21.

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

Example

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

Mathematica

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 *)

Prog

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

Crossrefs

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).

Sequence in context: A109551 A326393 A162278 * A027763 A233694 A338418

Adjacent sequences: A173924 A173925 A173926 * A173928 A173929 A173930

Keyword

nonn,more

Author

Michel Lagneau, Nov 26 2010

Extensions

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

Status

approved