cp's OEIS Frontend

A265788 Numerators of primes-only best approximates (POBAs) to sqrt(5); see Comments.

%I A265788 #10 Apr 06 2019 01:09:17
%S A265788 3,5,7,11,29,163,199,521,3571,26683,111667,150427,154841
%N A265788 Numerators of primes-only best approximates (POBAs) to sqrt(5); see Comments.
%C A265788 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 A265788 The POBAs to sqrt(5) start with 3/2, 5/2, 7/3, 11/5, 29/13, 163/73, 199/89, 521/233. For example, if p and q are primes and q > 89, then 199/89 is closer to sqrt(5) than p/q is.
%t A265788 x = Sqrt[5]; z = 1000; p[k_] := p[k] = Prime[k];
%t A265788 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265788 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265788 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265788 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265788 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265788 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265788 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265782/A265783 *)
%t A265788 Numerator[tL]   (* A265784 *)
%t A265788 Denominator[tL] (* A265785 *)
%t A265788 Numerator[tU]   (* A265786 *)
%t A265788 Denominator[tU] (* A265787 *)
%t A265788 Numerator[y]    (* A265788 *)
%t A265788 Denominator[y]  (* A265789 *)
%Y A265788 Cf. A000040, A265759, A265784, A265785, A265786, A265788, A265789.
%K A265788 nonn,frac,more
%O A265788 1,1
%A A265788 _Clark Kimberling_, Dec 26 2015
%E A265788 a(10)-a(13) from _Robert Price_, Apr 05 2019