A185816 Number of iterations of lambda(n) needed to reach 1.

0, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 3, 4, 3, 4, 3, 3, 4, 5, 2, 4, 3, 4, 3, 4, 3, 4, 3, 4, 4, 3, 3, 4, 4, 3, 3, 4, 3, 4, 4, 3, 5, 6, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 5, 3, 4, 4, 3, 4, 3, 4, 5, 4, 5, 3, 4, 3, 4, 4, 4, 4, 4, 3, 4, 3, 5, 4, 5, 3, 4, 4, 4

Offset

1,3

Comments

lambda(n) is the Carmichael lambda function, A002322.

a(n) = (length of row n in table A246700) - 1. - Reinhard Zumkeller, Sep 02 2014

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Paul Erdős, Andrew Granville, Carl Pomerance, and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.

Paul Erdős, Andrew Granville, Carl Pomerance, and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]

Paul Erdős, A. Granville, C. Pomerance, and C. Spiro, On the Normal Behavior of the Iterates of some Arithmetic Functions, in Analytic number theory (Allerton Park, IL, 1989), Progr. Math., 85 Birkhäuser Boston, Boston, MA, (1990), 165-204.

Nick Harland, The iterated Carmichael lambda function, arXiv:1111.3667v1 [math.NT], Nov 15, 2011.

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

Formula

For n > 1: a(n) = a(A002322(n)) + 1. - Reinhard Zumkeller, Sep 02 2014

Example

If n = 23 the trajectory is 23, 22, 10, 4, 2, 1. Its length is 6, thus a(23) = 5.

Maple

a:= n-> `if`(n=1, 0, 1+a(numtheory[lambda](n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 27 2019

Mathematica

f[n_] := Length[ NestWhileList[ CarmichaelLambda, n, Unequal, 2]] - 2; Table[f[n], {n, 1, 120}]

Prog

(Haskell)
a185816 n = if n == 1 then 0 else a185816 (a002322 n) + 1
-- Reinhard Zumkeller, Sep 02 2014

Crossrefs

Cf. A002322, A036459, A003434.

Cf. A027763, A173927, A246700.

Sequence in context: A350065 A077982 A331921 * A324901 A295568 A099427

Adjacent sequences: A185813 A185814 A185815 * A185817 A185818 A185819

Keyword

nonn

Author

Michel Lagneau, Feb 05 2011

Status

approved