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%I #10 Apr 05 2019 17:45:51
%S 5,3,5,19,71,601,1571,2579,3691,56813,111913
%N Numerators of primes-only best approximates (POBAs) to sqrt(3); 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 to sqrt(3) start with 5/2, 3/2, 5/3, 19/11, 71/41, 601/347, 1571/907. For example, if p and q are primes and q > 347, then 601/347 is closer to sqrt(3) than p/q is.
%t x = Sqrt[3]; z = 1000; 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, A265782/A265783 *)
%t Numerator[tL] (* A265778 *)
%t Denominator[tL] (* A265779 *)
%t Numerator[tU] (* A265780 *)
%t Denominator[tU] (* A265781 *)
%t Numerator[y] (* A265782 *)
%t Denominator[y] (* A265783 *)
%Y Cf. A000040, A265759, A265778, A265779, A265780, A265781, A265783.
%K nonn,frac,more
%O 1,1
%A _Clark Kimberling_, Dec 23 2015
%E a(10)-a(11) from _Robert Price_, Apr 05 2019
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