Gilles Bannay informed me of the following results he has found :
- For $k=4$, the solution $(2,3,14,60)$ is optimal for $(n_1,n_2,n_3,n_4) \leq (4,8,80,80)$.
- For $k=5$, the solution $(2,3,4,63,152)$ gives all numbers up to $450$.
- For $k=6$, the solution $(2,3,3,11,136,180)$ gives all numbers up to $2003$.
- Using the original rules from the French TV game, the $6$-uple 6$-tuple $(1,2,3,4,10,100)$ gives all numbers up to $1281$ (which answers my question in a comment).
He obtained these results using his program CEB, which can be downloaded here (the page is in English and contains a detailed explanation of all the options). He has added an option in order to search for the best $k$-tuples. For example, the result 1. above was found by typing :
> CEB -g -b1 -e10000 -a4 4 8 80 80
