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%I #23 Mar 26 2017 04:44:57
%S 0,0,0,0,0,0,1,0,0,0,1,1,2,3,0,0,4,1,3,2,0,0,0,3,1,2,2,3,2,0,1,0,0,1,
%T 3,0,2,3,17,1,2,1,16,2,0,0,18,2,19,1,2,2,17,1,15,3,1,2,15,1,19,20,15,
%U 0,4,18,14,19,17,16,21,2,20,21,30,22,29,16,27,3,24,25,14,19,22,23,14,20,23
%N Function f(x) = phi(x) + floor(x/2) is iterated; a(n) is the length of transient if the iteration was initiated at n.
%C Infinite iterations cannot be excluded, when a(n) is infinite. First at n=163?
%C For 1 <= n <= 1000, potentially infinite iterations at {163, 182, 196, 243, 283, 331, 423, 487, 495, 503, 511, 523, 533, 551, 559, 571, 583, 591, 593, 606, 611, 623, 642, 646, 651, 679, 685, 687, 725, 726, 729, 731, 732, 745, 746, 753, 755, 757, 758, 767, 779, 781, 783, 791, 799, 809, 811, 814, 839, 850, 855, 857, 859, 867, 869, 871, 875, 876, 885, 886, 888, 891, 895, 906, 908, 911, 913, 914, 915, 916, 921, 922, 923, 931, 937, 942, 959, 962, 964, 970, 971, 977, 985, 991} (tested to 1000 iterations). The maximum number of finite iterations in this range appears to be 96. - _Michael De Vlieger_, Mar 26 2017
%e n=70: iteration list = [70, 59, 87, 99, 109, 162, 135, 139, 207, 235, 301, 402, 333, 382, 381, 442, [413, 554, 553, 744, 612, 498], 413], so a(70)=16.
%e n=2^j: a(2^j)=0, powers of 2 are fixed points of f, free of transients.
%p a:= proc(n) local i, m, p;
%p p:= proc() -1 end; forget(p);
%p p(n):= 0; m:= n;
%p for i do m:= numtheory[phi](m)+iquo(m, 2);
%p if p(m)>-1 then return p(m) fi;
%p p(m):= i
%p od
%p end:
%p seq(a(n), n=1..162); # _Alois P. Heinz_, Nov 13 2015
%t Table[Count[Values@ PositionIndex@ NestList[EulerPhi@ # + Floor[#/2] &, n, 10^3], k_ /; Length@ k == 1], {n, 89}] (* _Michael De Vlieger_, Mar 26 2017, Version 10 *)
%Y Cf. A000010, A097026, A097028, A097029.
%K nonn
%O 1,13
%A _Labos Elemer_, Aug 27 2004
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