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A291785
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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.
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5
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.
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LINKS
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PROG
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(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
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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