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

1,5

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

Links

Table of n, a(n) for n=1..89.

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

Cf. A000010, A097026, A097028, A097029.

Sequence in context: A327759 A269978 A351258 * A289442 A092331 A299029

Adjacent sequences: A097025 A097026 A097027 * A097029 A097030 A097031

Keyword

nonn

Author

Labos Elemer, Aug 27 2004

Status

approved