Result similar to Gaussian Sums.

AtomSeven

Hi,

I think that I found a new result similar to Gaussian Sums. So expressing
N = p1^a1*p2^a2...pn^an, by the Fundamental Theorem of Arithmetic.

I found a closed form, f(n, pi), similar to a gaussian sum that for each of the pi expresses the corresponding ai=f(n, pi) .

So N = p1^(f(N, p1))*p2^(f(N,p2))*...*pn^(f(N,pn))


Does anyone knows of similar result?