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A265767 Numerators of upper primes-only best approximates (POBAs) to 5; see Comments.

%I A265767 #9 Mar 11 2017 21:13:12
%S A265767 11,37,67,97,157,307,337,367,397,487,547,757,787,907,967,997,1117,
%T A265767 1567,1657,1747,1867,1987,2287,2437,2617,2707,2857,2887,3037,3067,
%U A265767 3217,3307,3457,3547,3637,3697,3847,4057,4297,4597,4957,4987,5107,5167,5197,5347
%N A265767 Numerators of upper primes-only best approximates (POBAs) to 5; see Comments.
%C A265767 Suppose that x > 0. A fraction p/q of primes is an upper primes-only best approximate, and we write "p/q is in U(x)", if p'/q < x < p/q < u/v for all primes u and v such that v < q, where p' is greatest prime < p in case p >= 3.
%C A265767 Let q(1) = 2 and let p(1) be the least prime >= x. The sequence U(x) follows inductively: for n >= 1, let q(n) is the least prime q such that x < p/q < p(n)/q(n) for some prime p. Let q(n+1) = q and let p(n+1) be the least prime p such that x < p/q < p(n)/q(n).
%C A265767 For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.
%e A265767 The upper POBAs to 5 start with 11/2, 37/7, 67/13, 97/19, 157/31, 307/61, 337/67, 367/73. For example, if p and q are primes and q > 19, and p/q > 5, then 97/19 is closer to 5 than p/q is.
%t A265767 x = 5; z = 200; p[k_] := p[k] = Prime[k];
%t A265767 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265767 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265767 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265767 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265767 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265767 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265767 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265768/A265769 *)
%t A265767 Numerator[tL]   (* A265766 *)
%t A265767 Denominator[tL] (* A158318 *)
%t A265767 Numerator[tU]   (* A265767 *)
%t A265767 Denominator[tU] (* A023217 *)
%t A265767 Numerator[y]    (* A222568 *)
%t A265767 Denominator[y]  (* A265769 *)
%Y A265767 Cf. A000040, A023217, A158318, A265759, A265766, A222568, A265769.
%K A265767 nonn,frac
%O A265767 1,1
%A A265767 _Clark Kimberling_, Dec 19 2015