Convert a set of weighted real probabilities to a set of weighted integer odds

Question

What I have:

• A set of probabilities $$\{p_1, \dots , p_n\}$$ So that: $$\sum_{i=1}^n p_i = 1$$

What I want to obtain:

• An integer $$\alpha$$ and a set of integers $$\{o_1, ... , o_n\}$$ each mapped to the probability of respective index, so that $$\sum_{i=1}^n o_i/\alpha = 1$$

A concrete example:

1. Probabilities: 0.35, 0.30, 0.20, 0.10, 0.05
2. Integers: 7, 6, 4, 2, 1
3. Alpha: 20

I am in fact trying a way to generalize the above numbers :)

Thanks a lot in advance, hope everything is clear. I sense I might be talking about some math concept that I might not know about.