A291791 -id:A291791 - cp's OEIS frontend

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

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

A289997 Numbers n whose trajectory under iteration of the map k -> (sigma(k)+phi(k))/2 never reaches a fraction (that is, either the trajectory reaches a prime, which is a fixed point, or diverges to infinity).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 17, 19, 21, 22, 23, 26, 27, 29, 30, 31, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 51, 52, 53, 57, 58, 59, 60, 61, 65, 66, 67, 68, 71, 73, 74, 75, 79, 80, 82, 83, 89, 91, 92, 97, 101, 103, 106, 107, 109, 113, 114, 115, 116, 117, 126, 127, 131, 133, 134, 135, 136, 137

Offset: 1

Hugo Pfoertner, Sep 03 2017

nonn

Suggested by N. J. A. Sloane in a post "Iterating some number-theoretic functions" to the Seqfan mailing list.
The iteration arrives at a fixed point when k becomes a prime P, because sigma(P)=P+1 and phi(P)=P-1, hence k -> k.
It would be nice to have an independent characterization of these numbers (not involving the map in the definition). - N. J. A. Sloane, Sep 03 2017
Conjecturally, all terms of A291790 are in the sequence, because their trajectories (see example in A291789 for starting value 270) grow indefinitely. - Hugo Pfoertner, Sep 04 2017

			126 is in the sequence, because the following iteration arrives at a fixed point:
    k        sigma(k)     phi(k)
   126         312          36  k->(sigma(k)+phi(k))/2, (312+36)/2=174
   174         360          56  k->(sigma(k)+phi(k))/2, (360+56)/2=208
   208         434          96
   265         324         208
   266         480         108
   294         684          84
   384        1020         128
   574        1008         240
   624        1736         192
   964        1694         480
  1087        1088        1086  k->(sigma(k)+phi(k))/2, (1088+1086)/2=1087
  1087        1088        1086  ... loop

N. J. A. Sloane, Iterating some number-theoretic functions, Posting in Seqfan mailing list, Sep 3, 2017

Cf. A000203, A000010, A290001, A291789, A291790.

Complement of A291791.

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 *)

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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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A289997 Numbers n whose trajectory under iteration of the map k -> (sigma(k)+phi(k))/2 never reaches a fraction (that is, either the trajectory reaches a prime, which is a fixed point, or diverges to infinity).

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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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Mathematica