A289997 -id:A289997 - 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

A290001 Numbers k such that the iteration k -> (sigma(k)+phi(k))/2 terminates at a fraction later than at the first step.

Original entry on oeis.org

12, 14, 15, 20, 24, 28, 33, 34, 35, 42, 48, 54, 55, 56, 62, 63, 69, 70, 76, 77, 78, 84, 85, 86, 87, 88, 90, 93, 94, 95, 96, 99, 102, 104, 105, 108, 110, 111, 112, 118, 119, 120, 122, 123, 124, 125, 129, 130, 132, 141, 142, 143, 148, 150

Offset: 1

Hugo Pfoertner, Sep 03 2017

nonn

Termination at the first step occurs if k>3 is a square or twice a square, i.e. if k is a term of A028982. So the sequence lists the numbers that end at a fraction but are not of one of these two forms.

			42 is in the sequence because the following iteration reaches an odd sigma(k):
   k        sigma(k)     phi(k)
   42          96          12  k->(sigma(k)+phi(k))/2, (96+12)/2=54
   54         120          18  k->(sigma(k)+phi(k))/2, (120+18)/2=69
   69          96          44  ...
   70         144          24  ...
   84         224          24
  124         224          60
  142         216          70
  143         168         120  k->(sigma(k)+phi(k))/2, (168+120)/2=144
  144         403          48  sigma(144) is odd, iteration terminates

Hugo Pfoertner, Table of n, a(n) for n = 1..101
Hugo Pfoertner, Table of n, a(n) for n = 1..1136 (values above first 101 terms are only guesses)
N. J. A. Sloane, Iterating some number-theoretic functions, Posting to Seqfan mailing list, Sep 3, 2017
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, A028982, A289997, A291790, A291791.

b-file curtailed beyond last certain value, moved other values to a-file. - N. J. A. Sloane, Oct 01 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 *)

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.

A291791 Numbers n whose trajectory under the iteration of k -> (sigma(k)+phi(k))/2 reaches a fraction.

Original entry on oeis.org

4, 8, 9, 12, 14, 15, 16, 18, 20, 24, 25, 28, 32, 33, 34, 35, 36, 42, 48, 49, 50, 54, 55, 56, 62, 63, 64, 69, 70, 72, 76, 77, 78, 81, 84, 85, 86, 87, 88, 90, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111

Offset: 1

N. J. A. Sloane, Sep 03 2017

nonn

It would be nice to have an independent characterization of these numbers (not involving the map in the definition).

Hugo Pfoertner, Table of n, a(n) for n = 1..126

Union of A028982 \ {1,2} and A290001. Also the complement of A289997.

Cf. A000010, A000203.

B-file shortened by N. J. A. Sloane, Sep 17 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).

A291934 Largest k achievable by the iteration k->(sigma(k)+phi(k))/2 started at k=n, with termination when sigma(k) becomes odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 16, 13, 16, 16, 16, 17, 18, 19, 25, 23, 23, 23, 36, 25, 29, 29, 36, 29, 53, 31, 32, 36, 36, 36, 36, 37, 53, 53, 53, 41, 144, 43, 61, 61, 47, 47, 144, 49, 50, 61, 61, 53, 144, 72, 72, 59, 59, 59, 107, 61, 144

Offset: 1

Hugo Pfoertner, Sep 06 2017

nonn

See A291914 for types of termination. The first term with unknown conjecturally unbounded behavior is a(270). More values of n with this type of behavior are given in A291790.

All bounded terms of this sequence are either prime or in A028982.

			a(38)=53: Starting with k=38 produces the trajectory 38 -> 39 -> 50 -> 53 -> 53 ...
a(42)=144: See example in A290001.

Hugo Pfoertner, Table of n, a(n) for n = 1..269
Hugo Pfoertner, Illustration of sequence terms for n<=5000, omitting n in A291790.

CF. A000010, A000040, A000203, A028982, A289997, A290001, A291789, A291790, A291791, A291914.

With[{lim = 100}, TakeWhile[#, # > 0 &] &@ Table[Which[Length@ # < lim, Last@ #, SameQ @@ Take[#, -2], Last@ #, True, 0] &@ NestWhileList[(DivisorSigma[1, #] + EulerPhi[#])/2 &, n, EvenQ@ DivisorSigma[1, #] &, 1, lim], {n, 270}]] (* Michael De Vlieger, Sep 06 2017 *)

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.

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A290001 Numbers k such that the iteration k -> (sigma(k)+phi(k))/2 terminates at a fraction later than at the first step.

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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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A291791 Numbers n whose trajectory under the iteration of k -> (sigma(k)+phi(k))/2 reaches a fraction.

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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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A291934 Largest k achievable by the iteration k->(sigma(k)+phi(k))/2 started at k=n, with termination when sigma(k) becomes odd.

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