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A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

%I A265759 #19 Dec 29 2015 04:20:16
%S A265759 3,2,5,13,11,19,17,31,29,43,41,61,59,73,71,103,101,109,107,139,137,
%T A265759 151,149,181,179,193,191,199,197,229,227,241,239,271,269,283,281,313,
%U A265759 311,349,347,421,419,433,431,463,461,523,521,571,569,601,599,619,617
%N A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.
%C A265759 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. 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, ...).
%C A265759 See A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
%C A265759 x        Lower POBA       Upper POBA       POBA
%C A265759 1        A001359/A006512  A006512/A001359  A265759/A265760
%C A265759 3/2      A104163/A158708  A162336/A158709  A265761/A222565
%C A265759 2        A005383/A005382  A005385/A005384  A079149/A120628
%C A265759 3        A091180/A088878  A094525/A023208  A265763/A265764
%C A265759 4        A162857/A062737  A090866/A023212  A265765/A120639
%C A265759 5        A265766/A158318  A265767/A023217  A265768/A265769
%C A265759 6        A227756/A158015  A051644/A007693  A265770/A265771
%C A265759 sqrt(2)  A265772/A265773  A265774/A265775  A265776/A265777
%C A265759 sqrt(3)  A265778/A265779  A265780/A265781  A265782/A265783
%C A265759 sqrt(5)  A265784/A265785  A265786/A265787  A265788/A265789
%C A265759 sqrt(8)  A265790/A265791  A265792/A265793  A265794/A265795
%C A265759 tau      A265796/A265797  A265798/A265799  A265800/A265801
%C A265759 1/tau    A265799/A265798  A265797/A265796  A265806/A265807
%C A265759 pi       A265808/A265809  A265810/A265811  A265812/A265813
%C A265759 e        A265814/A265815  A265816/A265817  A265818/A265819
%e A265759 The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.
%t A265759 x = 1; z = 200; p[k_] := p[k] = Prime[k];
%t A265759 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265759 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265759 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265759 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265759 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265759 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265759 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265759/A265760 *)
%t A265759 Numerator[tL]   (* A001359 *)
%t A265759 Denominator[tL] (* A006512 *)
%t A265759 Numerator[tU]   (* A006512 *)
%t A265759 Denominator[tU] (* A001359 *)
%t A265759 Numerator[y]    (* A265759 *)
%t A265759 Denominator[y]  (* A265760 *)
%Y A265759 Cf. A000040, A001359, A006512, A265759, A265760.
%K A265759 nonn,frac
%O A265759 1,1
%A A265759 _Clark Kimberling_, Dec 15 2015