A185816 - OEIS

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

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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