maxpos[n_,p_,  α_, k_] := If[prop[n,k,p] ≤ α, maxpos[n,p,α, k+1], k-1]

prop[n_, k_, p_] := Sum[Binomial[n,l] p^l (1-p)^(n-l),{l,0,k}]

success[n_] := prop[n,maxpos[n, 1/1000,0.05,0],1/2000]

csuccess[n_] := Style[{n,success[n]},ColorData[3,"ColorList"][[Mod[maxpos[n,1/1000,0.05,0],10]]]]

ns = Table[100  1.06^k,{k,-0, 100}]

howmany[n_] := {n,maxpos[n, 1/1000, 0.05,0]}

ListPlot[howmany /@ ns]

ListPlot[csuccess /@ ns]