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

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

Author

Hugo Pfoertner, Sep 05 2017

Keywords

• sign

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

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

Crossrefs

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