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.

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

Author

Labos Elemer, Aug 27 2004

Keywords

• nonn

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

Examples

			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.
		

Crossrefs

Cf. A000010, A097026, A097028, A097029.

Programs

Mathematica

Table[Length@ Union@ NestList[EulerPhi@ # + Floor[#/2] &, n, 10^3], {n, 10^3}] (* Michael De Vlieger, Feb 27 2017 *)

Formula

a(n) = A097026(n) + A097027(n) = c(n) + t(n).