A180633 a(n) is the number of iterations of function f(x) = phi(x)-1 needed before zero is reached, when starting from the initial value x = n.

0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 3, 6, 4, 5, 5, 6, 4, 7, 5, 6, 5, 7, 5, 8, 6, 7, 6, 8, 5, 9, 6, 8, 6, 8, 6, 9, 7, 8, 6, 9, 6, 10, 8, 8, 7, 9, 6, 10, 8, 10, 8, 11, 7, 9, 8, 9, 8, 10, 6, 11, 9, 9, 10, 10, 8, 11, 10, 11, 8, 12, 8, 13, 9, 9, 9, 11, 8, 12, 10, 12, 9, 13, 8, 10, 10, 10, 9, 11, 8, 13, 11

Offset

0,4

Comments

Since phi(n) < n it follows that phi(n)-1 < n-1; therefore, after each iteration the argument decreases and eventually will reach zero.

Solution of equation phi(-1 + phi(-1 + phi(-1 + ...(phi(n))...))) = 1 where the totient function phi is applied a(n) times. (The original name of the sequence.)

Links

Antti Karttunen, Table of n, a(n) for n = 0..10000

Formula

a(0) = 0; for n >= 1, a(n) = 1 + a(A000010(n)-1). - Antti Karttunen, Aug 07 2017

Example

a(11)=5 since phi(-1+phi(-1+phi(-1+phi(-1+phi(-1+phi(11))))))=
phi(-1+phi(-1+phi(-1+phi(-1+phi(-1+10)))))=
phi(-1+phi(-1+phi(-1+phi(-1+6))))=
phi(-1+phi(-1+phi(-1+4)))=
phi(-1+phi(-1+2))=
phi(-1+1)=1 after 5 iterations.

Mathematica

f[n_] := If[n < 3, 1, Length@ NestWhileList[ EulerPhi@# -1 &, n, # != 1 &]]; Array[f, 93, 0] (* Robert G. Wilson v, Sep 25 2010 *)

Prog

(Scheme) (define (A180633 n) (if (zero? n) n (+ 1 (A180633 (+ -1 (A000010 n)))))) ;; Antti Karttunen, Aug 07 2017

Crossrefs

Cf. A000010.

Cf. A049108. - Robert G. Wilson v, Sep 25 2010

Sequence in context: A353862 A338719 A296080 * A286610 A335603 A304117

Adjacent sequences: A180630 A180631 A180632 * A180634 A180635 A180636

Keyword

nonn

Author

Carmine Suriano, Sep 13 2010

Extensions

Corrected a(18), a(19) & a(73) and extended past a(80) by Robert G. Wilson v, Sep 25 2010

Name changed and value of a(0) changed from 1 to 0 by Antti Karttunen, Aug 07 2017

Status

approved