cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A291804 a(n) = sigma(A291789(n)).

Original entry on oeis.org

720, 1092, 1224, 1350, 912, 2016, 2352, 3600, 3968, 4680, 7068, 9936, 11340, 13848, 18872, 28728, 43008, 59520, 101520, 124380, 175824, 189900, 311040, 433008, 618120, 747936, 1330560, 2219520, 3930240, 7607808, 15305472
Offset: 1

Views

Author

N. J. A. Sloane, Sep 07 2017

Keywords

Comments

Created in the hope that it will help explain the long-term behavior of A291789, which is the mean of this and A291805.

Links

Crossrefs

Cf. A291789, A291805.

A291805 a(n) = phi(A291789(n)).

Original entry on oeis.org

72, 120, 200, 352, 792, 280, 480, 464, 1008, 1240, 1152, 1088, 2496, 2304, 2688, 3360, 4560, 7920, 8960, 22080, 19520, 47040, 28640, 61440, 82400, 137760, 126336, 207360, 319488, 552960, 1013760, 2716416, 4505024, 7193472
Offset: 1

Views

Author

N. J. A. Sloane, Sep 07 2017

Keywords

Comments

Created in the hope that it will help explain the long-term behavior of A291789, which is the mean of this and A291804.

Links

Crossrefs

Cf. A291789, A291804.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Sep 03 2017

Keywords

Comments

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

Links

Crossrefs

Cf. A000010, A000203, A289997, A290001, A291789 (the trajectory of 270), A291787, A292108.
For the "seeds" see A292766.

Extensions

More terms from Hugo Pfoertner, Sep 03 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

Views

Author

Hugo Pfoertner, Sep 03 2017

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Views

Author

Hugo Pfoertner, Sep 05 2017

Keywords

Comments

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

Examples

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

References

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

Links

Crossrefs

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

Views

Author

Hugo Pfoertner, Sep 06 2017

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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