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A097028
Function f(x) = EulerPhi(x) + floor(x/2) is iterated; a(n) is the length of transient part and terminal cycle if the iteration was initiated at n. So a(n) is the number of distinct terms arising during iteration.
6
1, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 3, 3, 4, 1, 1, 5, 2, 5, 3, 2, 2, 4, 4, 2, 3, 4, 4, 6, 4, 3, 1, 4, 5, 5, 4, 4, 5, 23, 5, 4, 5, 22, 6, 2, 2, 24, 6, 25, 3, 3, 4, 23, 3, 21, 5, 2, 3, 21, 3, 25, 26, 21, 1, 6, 24, 20, 25, 23, 22, 27, 4, 26, 27, 36, 28, 35, 22, 33, 5, 30, 31, 20, 25, 28, 29, 20, 26, 29
OFFSET
1,5
COMMENTS
Observation (see also Labos Elemer's comment at A097026): most n < 1000 have 1 <= a(n) <= 108; the following have a(n) > 1000 if finite: {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}. - Michael De Vlieger, Feb 27 2017
FORMULA
a(n) = A097026(n) + A097027(n) = c(n) + t(n).
EXAMPLE
For 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}, a(70) = 22.
n=2^j: a(2^j)=1, powers of 2 are fixed points, free of transients, so t + c = 0 + 1 = 1.
MATHEMATICA
Table[Length@ Union@ NestList[EulerPhi@ # + Floor[#/2] &, n, 10^3], {n, 10^3}] (* Michael De Vlieger, Feb 27 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Aug 27 2004
STATUS
approved