

A291785


Iterate the map A291784: k > (psi(k)+phi(k))/2, starting with n, until a power of a prime (A000961) is reached, or 1 if that never happens.


5



1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 16, 13, 16, 16, 16, 17, 23, 19, 23, 23, 23, 23, 47, 25, 27, 27, 47, 29, 47, 31, 32, 83, 83, 83, 83, 37, 47, 47, 47, 41, 83, 43, 47, 1, 47, 47, 1, 49, 1, 83, 83, 53, 83, 1, 1, 59, 59, 59, 1, 61, 83, 83, 64, 83, 83, 67, 1, 1, 1, 71, 1, 73
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Primes and prime powers are fixed points under the map f(k) = (psi(k)+phi(k))/2. (If n = p^k, then psi(n) = p^k(1+1/p), phi(n) = p^k(11/p), and their average is p^k, so n is a fixed point under the map.)
Since f(n)>n if n is not a prime power, there can be no nontrivial cycles.
Wall (1985) observes that the trajectories of 45 and 50 are unbounded, so a(45) = a(50) = 1.
Also 48 and many more terms seem to have unbounded trajectories.  Hugo Pfoertner, Sep 03 2017.
Obviously any number in the trajectory of a number with unbounded trajectory (in particular that of 45, A291787) again has this property. A291788 is the union of all these.  M. F. Hasler, Sep 03 2017


REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.


LINKS

Table of n, a(n) for n=1..73.
C. R. Wall, Unbounded sequences of EulerDedekind means, Amer. Math. Monthly, 92 (1985), 587.


PROG

(PARI) A291785(n, L=n)={for(i=0, L, isprimepower(n=A291784(n))&&return(n)); (1)^(n>1)} \\ The search limit L=n is only experimental but appears quite conservative w.r.t. known data, cf. A291786. The algorithm assumes that there are no cycles except for the powers of primes.  M. F. Hasler, Sep 03 2017


CROSSREFS

Cf. A000010, A001615, A291784, A291786, A291787, A291788.
Sequence in context: A306369 A291784 A291934 * A122411 A339378 A341156
Adjacent sequences: A291782 A291783 A291784 * A291786 A291787 A291788


KEYWORD

sign


AUTHOR

N. J. A. Sloane, Sep 02 2017


EXTENSIONS

More terms from Hugo Pfoertner, Sep 03 2017


STATUS

approved



