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1. For $k=4$, the solution $(2,3,14,60)$ is optimal for $(n_1,n_2,n_3,n_4) \leq (4,8,80,80)$.
2. For $k=5$, the solution $(2,3,4,63,152)$ gives all numbers up to $450$.
3. For $k=6$, the solution $(2,3,3,11,136,180)$ gives all numbers up to $2003$.
4. 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  1 Gilles Bannay informed me of the following results he has found : 1. For$k=4$, the solution$(2,3,14,60)$is optimal for$(n_1,n_2,n_3,n_4) \leq (4,8,80,80)$. 2. For$k=5$, the solution$(2,3,4,63,152)$gives all numbers up to$450$. 3. For$k=6$, the solution$(2,3,3,11,136,180)$gives all numbers up to$2003$. 4. Using the original rules from the French TV game, the$6$-uple$(1,2,3,4,10,100)$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