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A265764 Denominators of primes-only best approximates (POBAs) to 3; see Comments.

%I A265764 #12 Jan 09 2016 14:23:15
%S A265764 2,2,5,5,7,7,11,13,13,17,19,23,23,29,37,37,43,43,47,53,59,61,67,71,79,
%T A265764 83,89,97,103,103,113,127,127,137,139,149,163,163,167,167,173,181,191,
%U A265764 193,197,199,211,227,233,239,251,257,257,263,269,271,277,293
%N A265764 Denominators of primes-only best approximates (POBAs) to 3; see Comments.
%C A265764 Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.
%e A265764 The POBAs for 3 start with  7/2, 5/2, 17/5, 13/5, 23/7, 19/7, 31/11, 41/13, 37/13, 53/17. For example, if p and q are primes and q > 13, then 41/13 is closer to 3 than p/q is.
%t A265764 x = 3; z = 200; p[k_] := p[k] = Prime[k];
%t A265764 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265764 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265764 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265764 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265764 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265764 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265764 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265763/A265764 *)
%t A265764 Numerator[tL]   (* A091180 *)
%t A265764 Denominator[tL] (* A088878 *)
%t A265764 Numerator[tU]   (* A094525 *)
%t A265764 Denominator[tU] (* A023208 *)
%t A265764 Numerator[y]    (* A265763 *)
%t A265764 Denominator[y]  (* A265764 *)
%Y A265764 Cf. A000040, A023208, A088878, A091180, A094525, A265763.
%K A265764 nonn,frac
%O A265764 1,1
%A A265764 _Clark Kimberling_, Dec 18 2015