Trying to find a simpler way to calculate this function

[SeventhSigma]

let P(n) = n^4 + an^3 + bn^2 + cn

M(a,b,c) returns largest m that divides P(n) for all n

then let function S(N) return the sum of all M(a,b,c) for 1 <= a,b,c <= N

I am trying to understand a simpler way to calculate S(N) so I don't have to actually process every single combination of a,b, and c but I am having trouble finding patterns to take advantage of on a broad scale.

So far I know from trying all sorts of values that M(a,b,c) tends to return values of form 2^i * 3^j where i,j>=0.

 

[mfb]

If m divides P(n) for all n, it also divides all differences: m divides P(2)-P(1) = 15+7a+3b, for example.
Using more values for n, you can eliminate more variables. This could help to reduce testing.

 

[coolul007]

P(n) can be looked at as a base 'n' number. This would make a,b,c digits and a max value for that part.