Behavior of the “mean prime factor” of numbers

This question concerns the behavior of
a function f() that maps each number in N to
its mean prime factor.
I previously posted premature questions, now deleted, which
explains the cites below to several who contributed observations.
Also, the question, “Distribution of the number of prime factors,”
may be relevant.

Define f(n) to be the floor of the geometric mean of
all the prime factors of n.
For example,
n=6500=225313=2255513.
f(6500)=(2255513)16=4.32=4.
For n a prime, f(n)=n, i.e., the primes are fixed points of f().
For n a prime power pk, f(n)=p (as Julian Rosen observed).
Greg Martin noted that
f(n)=n1/Ω(n)
where Ω(n) is the number of primes dividing
n counted with multiplicity.

I explored |f1(k)|, the number of nN
that map to k: f(n)=k.
The root transition points are crucial (as Gerhard Paseman emphasized).
For nmax, the square-root, cube-root, and fourth-roots
of n_\max are (3163,216,57) rounded up.
Here are graphs of |f^{-1}| at two scales:


         
Sqrt

         

n_\max=10^7. Note the transitions at 216 and 3163.


         
Cbrt

         

n_\max=10^7. Note the transitions at 57 and at 216.


Q.
What explains the apparent linearity in the first graph
in the range [216,3163], in contrast to the gradual
upsweep in the range [57,216] in the second graph?

I did not expect to see such regularity…

Answer

Attribution
Source : Link , Question Author : Joseph O’Rourke , Answer Author : Community

Leave a Comment