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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.

%I A180633 #20 Aug 07 2017 23:06:14
%S A180633 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,
%T A180633 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,
%U A180633 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
%N 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.
%C A180633 Since phi(n) < n it follows that phi(n)-1 < n-1; therefore, after each iteration the argument decreases and eventually will reach zero.
%C A180633 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.)
%H A180633 Antti Karttunen, <a href="/A180633/b180633.txt">Table of n, a(n) for n = 0..10000</a>
%F A180633 a(0) = 0; for n >= 1, a(n) = 1 + a(A000010(n)-1). - _Antti Karttunen_, Aug 07 2017
%e A180633 a(11)=5 since phi(-1+phi(-1+phi(-1+phi(-1+phi(-1+phi(11))))))=
%e A180633 phi(-1+phi(-1+phi(-1+phi(-1+phi(-1+10)))))=
%e A180633 phi(-1+phi(-1+phi(-1+phi(-1+6))))=
%e A180633 phi(-1+phi(-1+phi(-1+4)))=
%e A180633 phi(-1+phi(-1+2))=
%e A180633 phi(-1+1)=1 after 5 iterations.
%t A180633 f[n_] := If[n < 3, 1, Length@ NestWhileList[ EulerPhi@# -1 &, n, # != 1 &]]; Array[f, 93, 0] (* _Robert G. Wilson v_, Sep 25 2010 *)
%o A180633 (Scheme) (define (A180633 n) (if (zero? n) n (+ 1 (A180633 (+ -1 (A000010 n)))))) ;; _Antti Karttunen_, Aug 07 2017
%Y A180633 Cf. A000010.
%Y A180633 Cf. A049108. - _Robert G. Wilson v_, Sep 25 2010
%K A180633 nonn
%O A180633 0,4
%A A180633 _Carmine Suriano_, Sep 13 2010
%E A180633 Corrected a(18), a(19) & a(73) and extended past a(80) by _Robert G. Wilson v_, Sep 25 2010
%E A180633 Name changed and value of a(0) changed from 1 to 0 by _Antti Karttunen_, Aug 07 2017