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A265770 Numerators of primes-only best approximates (POBAs) to 6; see Comments.

%I A265770 #10 May 13 2017 16:53:25
%S A265770 13,11,19,17,31,29,43,41,67,79,103,101,113,139,137,173,223,257,283,
%T A265770 281,317,353,367,401,439,499,607,619,617,643,641,653,677,761,787,823,
%U A265770 821,907,941,977,1039,1087,1181,1193,1361,1373,1399,1433,1447,1543,1579
%N A265770 Numerators of primes-only best approximates (POBAs) to 6; see Comments.
%C A265770 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 A265770 The POBAs to 6 start with 13/2, 11/2, 19/3, 17/3, 31/5, 29/5, 43/7, 41/7, 67/11, 79/13, 103/17, 101/17. For example, if p and q are primes and q > 17, then 103/17 (and 101/17) is closer to 6 than p/q is.
%t A265770 x = 6; z = 200; p[k_] := p[k] = Prime[k];
%t A265770 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265770 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265770 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265770 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265770 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265770 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265770 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265770/A265771 *)
%t A265770 Numerator[tL]   (* A227756 *)
%t A265770 Denominator[tL] (* A158015 *)
%t A265770 Numerator[tU]   (* A051644 *)
%t A265770 Denominator[tU] (* A007693 *)
%t A265770 Numerator[y]    (* A222570 *)
%t A265770 Denominator[y]  (* A265771 *)
%Y A265770 Cf. A000040, A265759, A227756, A158015, A051644, A007693, A265771.
%K A265770 nonn,frac
%O A265770 1,1
%A A265770 _Clark Kimberling_, Dec 20 2015