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