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A291914 Termination behavior of the iteration k->(sigma(k)+phi(k))/2 when started at k=n. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

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

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

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

EXAMPLE

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

CROSSREFS

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

Sequence in context: A323090 A180026 A136176 * A303708 A319138 A330462

Adjacent sequences:  A291911 A291912 A291913 * A291915 A291916 A291917

KEYWORD

sign

AUTHOR

Hugo Pfoertner, Sep 05 2017

STATUS

approved

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Last modified April 21 13:10 EDT 2021. Contains 343153 sequences. (Running on oeis4.)