Suppose we have a positive integer n with a certain prime factorization.
Let n be almost arbitrary whole number.
Restrict n such that
n has prime factorization p0^b * p1 * p2 * ... * pq.
Let q = 0.
Then the count of the divisors of n is b.
Even simpler,
Let q = 0 and b = 0. Then n is a prime number.
We, include one as a factor and do not include n as a factor.
So all of n's 'factors for this purpose' are less than or equal to n/2.
So if n is a prime number, then the count of its 'factors' is one.
Define a function PFN(n) as 'proper factors of n'
~ Maple code for PFN(n) also called Divisors(n)
~ This may be hard to read for some.
Divisors := proc (n) local d, count; description " Enumerate all proper divisors. Assume a positive integer input. Then count the proper divisors."; print(" Input is ", n, " Begin calculation."); count := 0; for d to (1/2)*n do if `mod`(n, d) = 0 then count := count+1; print(" One proper divisor of ", n, " is the number ", d) end if end do; print(" and that is all of them. "); print(" count of proper divisors is ", count) end proc
~
So we can start a data table
n PFN(n)
2 1
3 1
5 1
7 1
now if q = 0 then n=p0^b
furthur, if b=2 then n is a square of a prime
and PFN(4) = 2.
similarly, for n as 3^2 then
we have PFN(3^2) = 2.
So the prime numbers can, in some sense be used interchangably
under this PFN(n) counting function.
More examples - PFN(5^2) = PFN(7^2) = 2.
mini conjecture by observation 1 - PFN( square of a prime number ) = 2.
similarly,
mini conjecture by observation 2 - PFN( cube of a prime number ) = 3.
as a check
PFN(3^9) = 9.
mini conjecture 3 - PFN ( [a prime number]^a ) = 1.
This is the first time I have typed this in.
example 17
PFN(11^17) calculation too long - waited 50 seconds.
PFN(3^13) is 13.
also, 3^13 = 1,594,323.
So, again, safe to say PFN(p^a) = a for any prime p and positive integer a.
My other, unexplained observations involve
Mersenne numbers Mc = 2^c - 1.
and a recursive doubling function
Given w(0) = a, as before in PFN(p^a) = a.
Then w(n) = 2*w(n-1) + 1.
Sure, as I type.
Needs more examples and a better writeup.
Away from keyboard. Done for now.
Matt