f(m,n) is a single algebraic function, not a program with if, elseif routines.

[QUOTE=a1call;462965]The truth table I gave in OP is false. Please disregard it.

Here are some actual results:

------------------------
m=7;n=11;
valuation(f(m,n),5)=0
------------------------
m=5;n=11;
valuation(f(m,n),5)=1
------------------------
m=7;n=5;
valuation(f(m,n),5)=0
------------------------
m=5;n=5;
valuation(f(m,n),5)=2
------------------------
m=25;n=5;
valuation(f(m,n),5)=3
------------------------
m=5;n=25;
valuation(f(m,n),5)=2
------------------------
m=7;n=25;
valuation(f(m,n),5)=0

So m is similar to an enable signal in digital electronics, and if enabled then n can contribute to valuation increase. But only as much as the valuation of m.[/QUOTE]

So maybe f(m, n) = prod_p p^(min(m, n) + m) where the product is over all primes p.

There are no Factorials. primorials, or Multifactorials in the function.

ETA There are also no int, floor, round or ceilings in the function. It is a classic algebraic function.

[QUOTE=a1call;462971]There are no Factorials. primorials, or Multifactorials in the function.

ETA There are also no int, floor, round or ceilings in the function. It is a classic [URL="https://en.wikipedia.org/wiki/Algebraic_function"]algebraic function[/URL].[/QUOTE]

what's the polynomial it's the root of ?

[QUOTE=science_man_88;462974]what's the polynomial it's the root of ?[/QUOTE]

Iff I understand it correctly the root is a function of n, but any of many different number of roots can be used.
This is assuming the root refers to the highest exponent in the function.

[QUOTE=a1call;462975]Iff I understand it correctly the root is a function of n, but any of many different number of roots can be used.
This is assuming the root refers to the highest exponent in the function.[/QUOTE]

a [URL="https://en.wikipedia.org/wiki/Zero_of_a_function"]root[/URL] is when a polynomial is 0.

[QUOTE=CRGreathouse;462970]So maybe f(m, n) = prod_p p^(min(m, n) + m) where the product is over all primes p.[/QUOTE]
Like I said there are no primorials in the function, but setting m as a primorial would be means of factoring n by GCD. But I am hoping for a more practical use.

[QUOTE=science_man_88;462976]a [URL="https://en.wikipedia.org/wiki/Zero_of_a_function"]root[/URL] is when a polynomial is 0.[/QUOTE]

Ok so solutions for m and n when f(m,n)=0
Scratch that.i don't know.i will let you know if I come up with a solution.

FYI I changed my last post.i don't know the root.

What would be the root of the function
2^x=0

[QUOTE=a1call;462980]What would be the root of the function
2^x=0[/QUOTE]

[TEX]2^{-\infty}\approx 0[/TEX] in theory . I may be wrong there though.