A291789 Trajectory of 270 under repeated application of k -> (phi(k)+sigma(k))/2.

270, 396, 606, 712, 851, 852, 1148, 1416, 2032, 2488, 2960, 4110, 5512, 6918, 8076, 10780, 16044, 23784, 33720, 55240, 73230, 97672, 118470, 169840, 247224, 350260, 442848, 728448, 1213440, 2124864, 4080384, 8159616, 13515078, 15767596, 18626016, 29239504, 39012864, 62623600, 92580308

Offset: 0

Author

N. J. A. Sloane, Sep 03 2017

Keywords

• nonn

Comments

The ultimate fate of this trajectory is presently unknown. It may reach a fractional value (when it dies), it may reach a prime (which would be a fixed point), it may enter a cycle of length greater than 1, or it may be unbounded. - Hugo Pfoertner and N. J. A. Sloane, Sep 18 2017

Links

• Sean A. Irvine, Table of n, a(n) for n = 0..515 [Terms through a(250) from Hugo Pfoertner, terms a(251)-a(356) from N. J. A. Sloane]
• Sean A. Irvine, Illustration of A291789 showing a(n+1)/a(n) (red), cumulative mean of a(n+1)/a(n) (green), and power of 2 in a(n) (blue)
• Hugo Pfoertner, Illustration of A291789 using a recursive 5th order Butterworth filter with normalized cut-off frequency of 0.1 (0.5<->Nyquist frequency) to smooth the data.
• N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 14.

Crossrefs

Cf. A000010, A000203, A289997, A290001, A291790, A291787, A291804, A291805.

See A291914 and A292108 for the "big picture".

Programs

Maple

orbit:= proc(n, m) uses numtheory;
  local V,k;
  V:= Vector(m);
  V[1]:= n;
  for k from 2 to m do V[k]:= (sigma(V[k-1])+ phi(V[k-1]))/2 od:
  convert(V,list)
end proc:
orbit(270, 200); # Robert Israel, Sep 07 2017

Mathematica

NestWhileList[If[! IntegerQ@ #, -1/2, (DivisorSigma[1, #] + EulerPhi@ #)/2] &, 270, Nor[! IntegerQ@ #, SameQ@ ##] &, 2, 38] (* Michael De Vlieger, Sep 19 2017 *)