This question concerns the behavior of

a function f() that maps each number in N to

itsmean 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=22⋅53⋅13=2⋅2⋅5⋅5⋅5⋅13.

f(6500)=⌊(2⋅2⋅5⋅5⋅5⋅13)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 |f−1(k)|, the number of n∈N

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:

^{ n_\max=10^7n_\max=10^7. Note the transitions at 216216 and 31633163. }

^{ n_\max=10^7n_\max=10^7. Note the transitions at 5757 and at 216216. }

.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*