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).

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

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: A276825 A270644 A361156 * A189225 A169988 A358052

Adjacent sequences: A097023 A097024 A097025 * A097027 A097028 A097029

Keyword

nonn

Author

Labos Elemer, Aug 27 2004

Status

approved