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%I #28 Sep 22 2017 11:58:31
%S 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,
%T 27,47,29,47,31,32,83,83,83,83,37,47,47,47,41,83,43,47,-1,47,47,-1,49,
%U -1,83,83,53,83,-1,-1,59,59,59,-1,61,83,83,64,83,83,67,-1,-1,-1,71,-1,73
%N 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.
%C 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.)
%C Since f(n)>n if n is not a prime power, there can be no nontrivial cycles.
%C Wall (1985) observes that the trajectories of 45 and 50 are unbounded, so a(45) = a(50) = -1.
%C Also 48 and many more terms seem to have unbounded trajectories. - _Hugo Pfoertner_, Sep 03 2017.
%C 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
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.
%H C. R. Wall, <a href="http://www.jstor.org/stable/2323173">Unbounded sequences of Euler-Dedekind means</a>, Amer. Math. Monthly, 92 (1985), 587.
%o (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
%Y Cf. A000010, A001615, A291784, A291786, A291787, A291788.
%K sign
%O 1,2
%A _N. J. A. Sloane_, Sep 02 2017
%E More terms from _Hugo Pfoertner_, Sep 03 2017
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