A291804 -id:A291804 - cp's OEIS frontend

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

N. J. A. Sloane, Sep 03 2017

nonn

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

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.

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

See A291914 and A292108 for the "big picture".

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

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

A292108 Iterate the map k -> (sigma(k) + phi(k))/2 starting at n; a(n) is the number of steps to reach either a fixed point or a fraction, or a(n) = -1 if neither of these two events occurs.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 3, 2, 1, 0, 1, 0, 2, 2, 1, 0, 4, 1, 2, 1, 4, 0, 2, 0, 1, 4, 3, 2, 1, 0, 3, 2, 1, 0, 9, 0, 2, 3, 1, 0, 7, 1, 1, 2, 1, 0, 8, 3, 2, 2, 1, 0, 3, 0, 8, 7, 1, 3, 2, 0, 1, 7, 6, 0, 1, 0, 3, 2, 4

Offset: 1

Hugo Pfoertner and N. J. A. Sloane, Sep 18 2017

nonn

The first unknown value is a(270).
For an alternative version of this sequence, see A291914.
From Andrew R. Booker, Sep 19 2017 and Oct 03 2017: (Start)
Let f(n) = (sigma(n) + phi(n))/2. Then f(n) >= n, so the trajectory of n under f either terminates with a half-integer, reaches a fixed point, or increases monotonically. The fixed points of f are 1 and the prime numbers, and f(n) is fractional iff n>2 is a square or twice a square.
It seems likely that a(n) = -1 for all but o(x) numbers n <= x. See link for details of the argument. (End)

			Let f(k) = (sigma(k) + phi(k))/2. Under the action of f:
14 -> 15 -> 16 -> 39/2, taking 3 steps, so a(14) = 3.
21 -> 22 -> 23, a prime, in 2 steps, so a(21) = 2.

Hugo Pfoertner, Table of n, a(n) for n = 1..269
Andrew R. Booker, Notes on (sigma + phi)/2
Sean A. Irvine, Showing how the initial portions of some of these trajectories merge
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.

Cf. A000010, A000203, A289997, A290001, A291790, A291787, A291804, A291805, A291914.

With[{i = 200}, Table[-1 + Length@ NestWhileList[If[! IntegerQ@ #, -1/2, (DivisorSigma[1, #] + EulerPhi@ #)/2] &, n, Nor[! IntegerQ@ #, SameQ@ ##] &, 2, i, -1] /. k_ /; k >= i - 1 -> -1, {n, 76}]] (* Michael De Vlieger, Sep 19 2017 *)

a(n) = 0 if n is 1 or a prime (these are fixed points).

a(n) = 1 if n>2 is a square or twice a square, since these reach a fraction in one step.

A291805 a(n) = phi(A291789(n)).

Original entry on oeis.org

72, 120, 200, 352, 792, 280, 480, 464, 1008, 1240, 1152, 1088, 2496, 2304, 2688, 3360, 4560, 7920, 8960, 22080, 19520, 47040, 28640, 61440, 82400, 137760, 126336, 207360, 319488, 552960, 1013760, 2716416, 4505024, 7193472

Offset: 1

N. J. A. Sloane, Sep 07 2017

nonn

Created in the hope that it will help explain the long-term behavior of A291789, which is the mean of this and A291804.

Sean A. Irvine, Table of n, a(n) for n = 1..504

Cf. A291789, A291804.

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

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A292108 Iterate the map k -> (sigma(k) + phi(k))/2 starting at n; a(n) is the number of steps to reach either a fixed point or a fraction, or a(n) = -1 if neither of these two events occurs.

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A291805 a(n) = phi(A291789(n)).

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