%I A007755
%S A007755 1,2,3,5,11,17,41,83,137,257,641,1097,2329,4369,10537,17477,35209,65537,
%T A007755 140417,281929,557057,1114129,2384897,4227137,8978569,16843009,35946497,
%U A007755 71304257,143163649,286331153,541073537,1086374209,2281701377,4295098369
%N A007755 Smallest number m such that the trajectory of m under iteration of Euler's 
               totient function phi(n) [A000010] contains exactly n distinct numbers 
               (including m and the fixed point).
%C A007755 Least integer k such that the number of iterations of Euler phi function 
               needed to reach 1 starting at k (k is counted) is n.
%C A007755 a(n) is smallest number in the class k(n) which groups families of integers 
               which take the same number of iterations of the totient function 
               to evolve to 1. The maximum is 2*3^(n-1).
%C A007755 Shapiro shows that the smallest number is greater the 2^(n-1). Catlin 
               shows that if a(n) is odd and composite, then its factors are among 
               the a(k), k < n. For example a(12) = a(5) a(8). There is a conjecture 
               that all terms of this sequence are odd. - T. D. Noe (noe(AT)sspectra.com), 
               Mar 08 2004
%C A007755 The indices of odd prime terms are given by n=A136040(k)+2 for k=1,2,
               3,.... - T. D. Noe, Dec 14 2007
%D A007755 P. A. Catlin, Concerning the iterated phi function, Amer. Math. Monthly, 
               Vol. 77, No. 1 (1970), 60-61.
%D A007755 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 29, Ellipses, 
               Paris 2008. Also Entry 137, p. 47.
%D A007755 R. K. Guy, Unsolved Problems in Number Theory, 2nd Ed. New York: Springer-Verlag, 
               p. 97, 1994, Section B41.
%D A007755 Harold Shapiro, An arithmetic function arising from the phi function, 
               Amer. Math. Monthly, Vol. 50, No. 1 (1943), 18-30.
%H A007755 T. D. Noe, <a href="b007755.txt">Table of n, a(n) for n=1..1002</a>
%H A007755 T. D. Noe, <a href="http://www.sspectra.com/math/IteratedPhi2.pdf">Computing 
               Numbers in Section I of the Totient Iteration</a> [From T. D. Noe 
               (noe(AT)sspectra.com), Nov 18 2008]
%F A007755 a(n+2) ~ 2^n.
%e A007755 n=3, a[3]=3 because trajectory={3,2,1}. n=1: a[1]=1 because trajectory={1}
%t A007755 f[n_] := Length[ NestWhileList[ EulerPhi, n, Unequal, 2]] - 1; a = Table[0, 
               {30}]; Do[b = f[n]; If[a[[b]] == 0, a[[b]] = n; Print[n, " = ", b]], 
               {n, 1, 22500000}] (from Robert G. Wilson v)
%Y A007755 Cf. A000010, A003434, A049108, A092873 (prime factors of a(n)), A060611, 
               A098196.
%Y A007755 Sequence in context: A062737 A085613 A082605 this_sequence A060611 A103598 
               A077497
%Y A007755 Adjacent sequences: A007752 A007753 A007754 this_sequence A007756 A007757 
               A007758
%K A007755 nonn
%O A007755 1,2
%A A007755 Pepijn van Erp [ vanerp(AT)sci.kun.nl ]
%E A007755 More terms from David W. Wilson (davidwwilson(AT)comcast.net) May 15 
               1997
%E A007755 Additional comments from James S. Cronen (cronej(AT)rpi.edu)