x = Sqrt[3]; z = 1000; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], 1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1  x] > Abs[#2  x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, #  1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265782/A265783 *)
Numerator[tL] (* A265778 *)
Denominator[tL] (* A265779 *)
Numerator[tU] (* A265780 *)
Denominator[tU] (* A265781 *)
Numerator[y] (* A262582 *)
Denominator[y] (* A265783 *)
