How to formally write $f\left(k\right)={1\over p_1}+{1\over p_1p_2}+{1\over p_1^2p_2p_3}+{1\over p_1^4p_2^2p_3p_4}+\dots$

Question

How do I write the following finite series as a sum of products:

$$f\left( k \right) = {1 \over p_1} + {1 \over p_1p_2} + {1 \over p_1^2p_2p_3} + {1 \over p_1^4p_2^2p_3p_4} + \dots + {1 \over p_1^{2^{k-2}}p_2^{2^{k-3}} \dots p_{k-3}^4p_{k-2}^2p_{k-1}p_k}$$

This is somewhat similar to my previous question.

Answer 1

Using the Sigma - Pi notation:

$$f(k) = \sum\limits_{n=1}^{k}\left(\dfrac{1}{p_n}\prod\limits_{m=1}^{n-1}\dfrac{1}{p_m^{2^{n-1-m}}}\right)$$