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A265768 Numerators of primes-only best approximates (POBAs) to 5; see Comments.

%I A265768 #6 Dec 20 2015 13:51:04
%S A265768 7,11,23,37,53,67,83,97,113,157,233,263,293,307,337,353,367,397,443,
%T A265768 487,503,547,563,653,683,743,757,787,863,907,953,967,983,997,1117,
%U A265768 1163,1193,1283,1553,1567,1583,1657,1733,1747,1867,1913,1987,2003,2153,2213
%N A265768 Numerators of primes-only best approximates (POBAs) to 5; see Comments.
%C A265768 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 A265768 The POBAs to 5 start with 7/2, 11/2, 23/5, 37/7, 53/11, 67/13, 83/17, 97/19, 113/23, 157/31, 233/47. For example, if p and q are primes and q > 13, then 67/13 is closer to 5 than p/q is.
%t A265768 x = 5; z = 200; p[k_] := p[k] = Prime[k];
%t A265768 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265768 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265768 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265768 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265768 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265768 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265768 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265768/A265769 *)
%t A265768 Numerator[tL]   (* A265766 *)
%t A265768 Denominator[tL] (* A158318 *)
%t A265768 Numerator[tU]   (* A265767 *)
%t A265768 Denominator[tU] (* A023217 *)
%t A265768 Numerator[y]    (* A222568 *)
%t A265768 Denominator[y]  (* A265769 *)
%Y A265768 Cf. A000040, A023217, A158318, A265759, A265766, A265767, A265769.
%K A265768 nonn,frac
%O A265768 1,1
%A A265768 _Clark Kimberling_, Dec 19 2015