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.

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

1,12

Comments

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)

Links

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.

Formula

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.

Example

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.

Mathematica

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 *)

Crossrefs

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

Sequence in context: A271868 A194354 A156776 * A352910 A362328 A342176

Adjacent sequences: A292105 A292106 A292107 * A292109 A292110 A292111

Keyword

nonn

Author

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

Status

approved