A292108 -id:A292108 - cp's OEIS frontend

A291790 Numbers whose trajectory under iteration of the map k -> (sigma(k)+phi(k))/2 consists only of integers and is unbounded.

270, 290, 308, 326, 327, 328, 352, 369, 393, 394, 395, 396, 410, 440, 458, 459, 465, 496, 504, 510, 525, 559, 560, 570, 606, 616, 620, 685, 686, 702, 712, 725, 734, 735, 737, 738, 745, 746, 747, 783, 791, 792, 805, 806, 813, 814, 815, 816, 828

Offset: 1

N. J. A. Sloane, Sep 03 2017

nonn

It would be nice to have a proof that these trajectories are integral and unbounded, or, of course, that they eventually reach a fractional value (and die), or reach a prime (which is then a fixed point). (Cf. A291787.) If either of the last two things happen, then that value of n will be removed from the sequence. AT PRESENT ALL TERMS ARE CONJECTURAL.

When this sequence was submitted, there was a hope that it would be possible to prove that these trajectories were indeed integral and unbounded. This has not yet happened, although see the remarks of Andrew R. Booker in A292108. - N. J. A. Sloane, Sep 25 2017

Hugo Pfoertner, Table of n, a(n) for n = 1..82
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)

Cf. A000010, A000203, A289997, A290001, A291789 (the trajectory of 270), A291787, A292108.

For the "seeds" see A292766.

More terms from Hugo Pfoertner, Sep 03 2017

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

Original entry on oeis.org

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 *)

A291914 Termination behavior of the iteration k->(sigma(k)+phi(k))/2 when started at k=n.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Sep 05 2017

sign

The sequence tries to combine all possible cases, using the following definitions:
  - a(n) = 0 if n>2 is a square or twice a square, i.e. if n is in A028982\{1,2};
  - otherwise, a(n) = -1 if n is a prime P, because the trajectory immediately enters the loop of length 1 (sigma(P)+phi(P))/2=P (i.e. if n in A000040);
  - otherwise, a(n) = number of steps (>1) to fracture, i.e. when sigma(k) becomes odd and the iteration dies (n in A290001);
  - otherwise, a(n) = negative of number of steps to k becoming a prime at which point the trajectory has reached a fixed point and loops (n in A289997);
  - otherwise a(n) = 200 if the trajectory has grown for at least 200 steps without fracturing or running into a loop (n in A291790).
This is somewhat unsatisfactory, since it "depends on an arbitrary but large parameter", namely 200.  Once this sequence is better understood, the last clause can either be replaced by something like "a(n) = 9999999999999999 if the trajectory increases without limit" or simply omitted if it can be proved that case never happens. See A292108 for another version of this sequence. - N. J. A. Sloane, Sep 05 2017

			See examples in A289997 demonstrating a(126)=-11, and in A290001 demonstrating a(42)=9.

Richard K. Guy, Unsolved Problems in Number Theory, Third Edition, Springer, New York, 2004. Section B41, Iterations of phi and sigma, p. 147.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000203, A289997, A290001, A291789, A291790, A291791, A291934, A292108 (another version).

A292766 Numbers n whose trajectory under iteration of the map k -> (sigma(k)+phi(k))/2 consists only of integers and is unbounded, excluding numbers n whose trajectory merges with the trajectory of a smaller number.

Original entry on oeis.org

270, 440, 496, 702, 737, 813, 828, 897, 905, 1027, 1066, 1099, 1240, 1241, 1260, 1331, 1353, 1368, 1371, 1422, 1507, 1537, 1754, 1760, 1834, 1848, 2002, 2016, 2282

Offset: 1

N. J. A. Sloane, Sep 27 2017, based on emails from Sean A. Irvine, Sep 14 2017, who computed a(1)-a(9), and Hans Havermann, same date, who computed a(10)-a(29). Hugo Pfoertner also computed many of these terms

nonn

These are the "seeds" in A291790, that is, every number which blows up under iteration of the map k -> (sigma(k)+phi(k))/2 belongs to one of these trajectories. AT PRESENT ALL TERMS ARE CONJECTURAL.
The trajectories of these numbers are pairwise disjoint for the first 400 steps.
This is unsatisfactory because it is possible that, at some later step, these trajectories may merge, reach a prime (a fixed point), or reach a fraction (and die). However, this seems unlikely on probabilistic grounds - see the remarks of Andrew R. Booker in A292108.
Normally such a sequence would not be included in the OEIS, but exceptions have been made for this and A291790 because a number of people have worked on them, and also in the hope that this will encourage resolution of some of the open questions.
Needs a b-file.

Sean A. Irvine, Showing how the initial portions of some of these trajectories merge

Cf. A291790, A292108.

A292770 If sigma(n)+phi(n) is even then a(n) = (sigma(n)+phi(n))/2 otherwise -1.

Original entry on oeis.org

1, 2, 3, -1, 5, 7, 7, -1, -1, 11, 11, 16, 13, 15, 16, -1, 17, -1, 19, 25, 22, 23, 23, 34, -1, 27, 29, 34, 29, 40, 31, -1, 34, 35, 36, -1, 37, 39, 40, 53, 41, 54, 43, 52, 51, 47, 47, 70, -1, -1, 52, 61, 53, 69, 56, 72, 58, 59, 59, 92, 61, 63, 70, -1, 66, 82, 67, 79, 70, 84, 71, -1, 73, 75, 82, 88, 78, 96

Offset: 1

N. J. A. Sloane, Sep 28 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A028982, A065387, A291790, A292108, A292766, A292771.

f:= proc(n) local v;
   v:= numtheory:-sigma(n)+numtheory:-phi(n);
   if v::even then v/2 else -1 fi
end proc:
map(f, [$1..100]); # Robert Israel, Sep 28 2017

Table[If[EvenQ[DivisorSigma[1,n]+EulerPhi[n]],(DivisorSigma[1,n]+EulerPhi[n])/2,-1],{n,120}] (* Harvey P. Dale, Jun 19 2022 *)

a(n) = {my(f = factor(n), s = sigma(f) + eulerphi(f)); if(s%2, -1, s/2);} \\ Amiram Eldar, Dec 06 2024

a(n) = -1 if and only if n is in A028982 \ {1, 2}. - Amiram Eldar, Dec 06 2024

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A291790 Numbers whose trajectory under iteration of the map k -> (sigma(k)+phi(k))/2 consists only of integers and is unbounded.

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

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A291914 Termination behavior of the iteration k->(sigma(k)+phi(k))/2 when started at k=n.

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A292766 Numbers n whose trajectory under iteration of the map k -> (sigma(k)+phi(k))/2 consists only of integers and is unbounded, excluding numbers n whose trajectory merges with the trajectory of a smaller number.

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A292770 If sigma(n)+phi(n) is even then a(n) = (sigma(n)+phi(n))/2 otherwise -1.

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