Exponential function in which all solutions add up to a certain number

Question

I have been working on an function for a program I am writing and have hit a road block when getting down to creating said function, so I am asking for help here. I want to create a function (or a combination of functions) that:

1. is exponential
2. whose domain is capped at positive whole number r, a variable that is set before the function is calculated.
3. all positive whole number solutions of the function, between 1 and r (counting both 1 and r), add up to 100,000.
4. at the x value of 1, the function returns zero.

This question has stumped me for the last six days, so I would greatly appreciate the help!