A246700 -id:A246700 - cp's OEIS frontend

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

Michel Lagneau, Feb 05 2011

nonn

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

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

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

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.

Cf. A002322, A036459, A003434.

Cf. A027763, A173927, A246700.

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

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

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

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

A375478 Irregular triangle read by rows in which row n lists the iterates of the phi(x) map from n to 1, where phi(x) is Euler's totient function (A000010).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 2, 1, 5, 4, 2, 1, 6, 2, 1, 7, 6, 2, 1, 8, 4, 2, 1, 9, 6, 2, 1, 10, 4, 2, 1, 11, 10, 4, 2, 1, 12, 4, 2, 1, 13, 12, 4, 2, 1, 14, 6, 2, 1, 15, 8, 4, 2, 1, 16, 8, 4, 2, 1, 17, 16, 8, 4, 2, 1, 18, 6, 2, 1, 19, 18, 6, 2, 1, 20, 8, 4, 2, 1, 21, 12, 4, 2, 1

Offset: 1

Paolo Xausa, Aug 17 2024

First differs from A246700 at n = 22.

			Triangle begins:
   1;
   2, 1;
   3, 2, 1;
   4, 2, 1;
   5, 4, 2, 1;
   6, 2, 1;
   7, 6, 2, 1;
   8, 4, 2, 1;
   9, 6, 2, 1;
  10, 4, 2, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10636 (rows 1..1200 of the triangle, flattened).
Index entries for sequences related to totient function phi(n).

Supersequence of A246700.

Cf. A000010, A049108 (row lengths), A053478 (row sums).

Array[Most[FixedPointList[EulerPhi, #]] &, 25]

T(n,1) = n; T(n,k) = A000010(T(n,k-1)), for k = 2..A049108(n).

cp's OEIS Frontend

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

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