A291786 a(n) = number of iterations of k -> (psi(k)+phi(k))/2 (A291784) needed to reach a prime or a power of a prime or 1, or -1 if that doesn't happen.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0, 2, 1, 0, 0, 3, 0, 2, 2, 1, 0, 6, 0, 1, 0, 5, 0, 4, 0, 0, 9, 8, 7, 6, 0, 5, 4, 3, 0, 5, 0, 2, -1, 1, 0, -1, 0, -1, 6, 5, 0, 4, -1, -1, 2, 1, 0, -1, 0, 4, 3, 0, 3, 2, 0, -1, -1, -1, 0, -1, 0

Offset

1,12

Comments

Primes and prime powers are fixed points under the map f(k) = (psi(k)+phi(k))/2, so in that case we take a(n)=0. (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. See A291787, A291788.

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 Euler-Dedekind means, Amer. Math. Monthly, 92 (1985), 587.

Formula

a(n) = 0 iff n is in A000961. - M. F. Hasler, Sep 03 2017

Prog

(PARI) A291786(n, L=n)=n>1&&for(i=0, L, isprimepower(n)&&return(i); n=A291784(n)); -(n>1) \\ The suggested search limit L=n is only empirical and might require revision. The code also currently assumes that the prime powers are the only cycles. - M. F. Hasler, Sep 03 2017

Crossrefs

Cf. A000010, A001615, A291784, A291785, A291787, A291788.

Sequence in context: A072780 A124452 A351532 * A004603 A174951 A275326

Adjacent sequences: A291783 A291784 A291785 * A291787 A291788 A291789

Keyword

sign,more

Author

N. J. A. Sloane, Sep 02 2017

Extensions

Initial terms corrected and more terms from M. F. Hasler, Sep 03 2017

Status

approved