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(1-1/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
C. R. Wall, Unbounded sequences of Euler-Dedekind 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
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Sep 02 2017
EXTENSIONS
More terms from Hugo Pfoertner, Sep 03 2017
STATUS
approved