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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.

%I A292108 #48 Jan 14 2023 09:49:50
%S A292108 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,
%T A292108 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,
%U A292108 7,6,0,1,0,3,2,4
%N 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.
%C A292108 The first unknown value is a(270).
%C A292108 For an alternative version of this sequence, see A291914.
%C A292108 From _Andrew R. Booker_, Sep 19 2017 and Oct 03 2017: (Start)
%C A292108 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.
%C A292108 It seems likely that a(n) = -1 for all but o(x) numbers n <= x. See link for details of the argument. (End)
%H A292108 Hugo Pfoertner, <a href="/A292108/b292108.txt">Table of n, a(n) for n = 1..269</a>
%H A292108 Andrew R. Booker, <a href="/A292108/a292108.pdf">Notes on (sigma + phi)/2</a>
%H A292108 Sean A. Irvine, <a href="/A291790/a291790.png">Showing how the initial portions of some of these trajectories merge</a>
%H A292108 N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, <a href="https://vimeo.com/237029685">Part I</a>, <a href="https://vimeo.com/237030304">Part 2</a>, <a href="https://oeis.org/A290447/a290447_slides.pdf">Slides.</a> (Mentions this sequence)
%H A292108 N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 14.
%F A292108 a(n) = 0 if n is 1 or a prime (these are fixed points).
%F A292108 a(n) = 1 if n>2 is a square or twice a square, since these reach a fraction in one step.
%e A292108 Let f(k) = (sigma(k) + phi(k))/2. Under the action of f:
%e A292108 14 -> 15 -> 16 -> 39/2, taking 3 steps, so a(14) = 3.
%e A292108 21 -> 22 -> 23, a prime, in 2 steps, so a(21) = 2.
%t A292108 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 *)
%Y A292108 Cf. A000010, A000203, A289997, A290001, A291790, A291787, A291804, A291805, A291914.
%K A292108 nonn
%O A292108 1,12
%A A292108 _Hugo Pfoertner_ and _N. J. A. Sloane_, Sep 18 2017