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%I #19 Dec 29 2015 04:20:16
%S 3,2,5,13,11,19,17,31,29,43,41,61,59,73,71,103,101,109,107,139,137,
%T 151,149,181,179,193,191,199,197,229,227,241,239,271,269,283,281,313,
%U 311,349,347,421,419,433,431,463,461,523,521,571,569,601,599,619,617
%N Numerators of primes-only best approximates (POBAs) to 1; 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. 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 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 x Lower POBA Upper POBA POBA
%C 1 A001359/A006512 A006512/A001359 A265759/A265760
%C 3/2 A104163/A158708 A162336/A158709 A265761/A222565
%C 2 A005383/A005382 A005385/A005384 A079149/A120628
%C 3 A091180/A088878 A094525/A023208 A265763/A265764
%C 4 A162857/A062737 A090866/A023212 A265765/A120639
%C 5 A265766/A158318 A265767/A023217 A265768/A265769
%C 6 A227756/A158015 A051644/A007693 A265770/A265771
%C sqrt(2) A265772/A265773 A265774/A265775 A265776/A265777
%C sqrt(3) A265778/A265779 A265780/A265781 A265782/A265783
%C sqrt(5) A265784/A265785 A265786/A265787 A265788/A265789
%C sqrt(8) A265790/A265791 A265792/A265793 A265794/A265795
%C tau A265796/A265797 A265798/A265799 A265800/A265801
%C 1/tau A265799/A265798 A265797/A265796 A265806/A265807
%C pi A265808/A265809 A265810/A265811 A265812/A265813
%C e A265814/A265815 A265816/A265817 A265818/A265819
%e 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 x = 1; 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, A265759/A265760 *)
%t Numerator[tL] (* A001359 *)
%t Denominator[tL] (* A006512 *)
%t Numerator[tU] (* A006512 *)
%t Denominator[tU] (* A001359 *)
%t Numerator[y] (* A265759 *)
%t Denominator[y] (* A265760 *)
%Y Cf. A000040, A001359, A006512, A265759, A265760.
%K nonn,frac
%O 1,1
%A _Clark Kimberling_, Dec 15 2015
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