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A265761 Numerators of primes-only best approximates (POBAs) to 3/2; see Comments.

%I A265761 #12 Jan 09 2016 14:22:57
%S A265761 2,5,7,11,17,19,29,43,47,61,71,79,89,101,107,109,151,163,191,197,223,
%T A265761 227,251,269,271,317,349,359,421,439,461,467,521,523,569,601,613,631,
%U A265761 647,659,673,691,701,719,811,821,853,857,881,911,919,929,947,971,991
%N A265761 Numerators of primes-only best approximates (POBAs) to 3/2; see Comments.
%C A265761 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 A265761 The POBAs for 3/2 start with 2/2, 5/3, 7/5, 11/7, 17/11, 19/13, 29/19, 43/29, 47/31. For example, if p and q are primes and q > 13, then 19/13 is closer to 3/2 than p/q is.
%t A265761 x = 3/2; z = 200; p[k_] := p[k] = Prime[k];
%t A265761 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265761 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265761 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265761 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265761 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265761 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265761 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265761/A222565 *)
%t A265761 Numerator[tL]   (* A104163 *)
%t A265761 Denominator[tL] (* A158708 *)
%t A265761 Numerator[tU]   (* A162336 *)
%t A265761 Denominator[tU] (* A158709 *)
%t A265761 Numerator[y]    (* A265761 *)
%t A265761 Denominator[y]  (* A222565 *)
%Y A265761 Cf. A000040, A104163, A158708, A158790, A162336, A222565.
%K A265761 nonn,frac
%O A265761 1,1
%A A265761 _Clark Kimberling_, Dec 18 2015