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A265766 Numerators of lower primes-only best approximates (POBAs) to 5; see Comments.

%I A265766 #9 Mar 11 2017 21:13:05
%S A265766 7,13,23,53,83,113,233,263,293,353,443,503,563,653,683,743,863,953,
%T A265766 983,1163,1193,1283,1553,1583,1733,1913,2003,2153,2213,2243,2333,2393,
%U A265766 2543,2843,2963,3083,3203,3413,3593,3803,3863,4133,4283,4643,4703,4733,5153
%N A265766 Numerators of lower primes-only best approximates (POBAs) to 5; see Comments.
%C A265766 Suppose that x > 0. A fraction p/q of primes is a lower primes-only best approximate, and we write "p/q is in L(x)", if u/v < p/q < x < p'/q for all primes u and v such that v < q, where p' is least prime > p.
%C A265766 Let q(1) be the least prime q such that u/q < x for some prime u, and let p(1) be the greatest such u. The sequence L(x) follows inductively: for n > 1, let q(n) is the least prime q such that p(n)/q(n) < p/q < x for some prime p. Let q(n+1) = q and let p(n+1) be the greatest prime p such that p(n)/q(n) < p/q < x.
%C A265766 For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.
%e A265766 The lower POBAs to 5 start with 7/2, 13/3, 23/5, 53/11, 83/17, 113/23, 233/47. For example, if p and q are primes and q > 17, and p/q < 5, then 83/17 is closer to 5 than p/q is.
%t A265766 x = 5; z = 200; p[k_] := p[k] = Prime[k];
%t A265766 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265766 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265766 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265766 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265766 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265766 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265766 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265768/A265769 *)
%t A265766 Numerator[tL]   (* A265766 *)
%t A265766 Denominator[tL] (* A158318 *)
%t A265766 Numerator[tU]   (* A265767 *)
%t A265766 Denominator[tU] (* A023217 *)
%t A265766 Numerator[y]    (* A222568 *)
%t A265766 Denominator[y]  (* A265769 *)
%Y A265766 Cf. A000040, A023217, A158318, A222568, A265759, A265767, A265769.
%K A265766 nonn,frac
%O A265766 1,1
%A A265766 _Clark Kimberling_, Dec 19 2015