

A097026


Function f(x) = phi(x) + floor(x/2) is iterated, starting at x=n; a(n) is the length of terminal cycle (or 0 if no finite cycle exists).


6



1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1, 1, 2, 1, 4, 4, 2, 1, 4, 4, 2, 4, 2, 2, 6, 4, 2, 4, 6, 4, 2, 2, 6, 4, 6, 2, 1, 2, 6, 2, 6, 2, 1, 1, 6, 2, 6, 6, 6, 1, 2, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6
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OFFSET

1,5


COMMENTS

While iteration of phi(x) always leads to a fixed point, f(x) = phi(x) + incr(x) may result in cycles or be divergent. What is the magnitude of the added incrementing function?
Some initial values are hard to analyze. The first is n=163, so perhaps a(163)=0 by definition.
Observation regarding the above comment: most n <= 1000 have 1 <= a(n) <= 12; the following have unresolved cycles at 10^3 iterations of f(x): {163, 182, 196, 243, 283, 331, 423, 487, 495, 503, 511, 523, 533, 551, 559, 571, 583, 591, 593, ...}.  Michael De Vlieger, May 16 2017


LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..162
Michael De Vlieger, Table of n, a(n) for n = 1..1000 with 0 representing terms that have unresolved cycles at 10^3 iterations of f(x).


FORMULA

For n=2^j: a(2^j)=1, powers of 2 are fixed points.


EXAMPLE

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)=6;


MATHEMATICA

With[{nn = 10^3}, Table[Count[Values@ PositionIndex@ NestList[EulerPhi@ # + Floor[#/2] &, n, nn], s_ /; Length@ s > 1], {n, 105}]] (* Michael De Vlieger, May 16 2017 *)


PROG

(PARI) findpos(newn, v) = {forstep(k=#v, 1, 1, if (v[k] == newn, return(k)); ); }
a(n) = {ok = 0; v = [n]; while(!ok, newn = eulerphi(n) + n\2; ipos = findpos(newn, v); if (ipos, ok = 1; break); v = concat(v, newn); n = newn; ); #v  ipos + 1; } \\ Michel Marcus, Jan 03 2017


CROSSREFS

Cf. A000010, A097027, A097028, A097029.
Sequence in context: A144703 A276825 A270644 * A189225 A169988 A213259
Adjacent sequences: A097023 A097024 A097025 * A097027 A097028 A097029


KEYWORD

nonn


AUTHOR

Labos Elemer, Aug 27 2004


STATUS

approved



