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A265812 Numerators of primes-only best approximates (POBAs) to Pi; see Comments.

%I A265812 #17 Apr 29 2025 13:38:59
%S A265812 5,7,17,23,41,167,211,223,619,757,977,1109,4483,5237,5413,9497,14423,
%T A265812 16063,18061,30841,45751,47881,60661,137341,162901,177811,211891,
%U A265812 626443,833719,38144863,40436969,45230587,93379723,114431749,120059441,185091653,347672183,1725229397,1736068099
%N A265812 Numerators of primes-only best approximates (POBAs) to Pi; see Comments.
%C A265812 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. Many terms of A265806 are also terms of A265801 (denominators of POBAs to tau).
%e A265812 The POBAs to Pi start with 5/2, 7/2, 17/5, 23/7, 41/13, 167/53, 211/67, 223/71, 619/197. For example, if p and q are primes and q < 53, then 167/53 is closer to Pi than p/q is.
%t A265812 x = Pi; z = 1000; p[k_] := p[k] = Prime[k];
%t A265812 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265812 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265812 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265812 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265812 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265812 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265812 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265812/A265813 *)
%t A265812 Numerator[tL]   (* A265808 *)
%t A265812 Denominator[tL] (* A265809 *)
%t A265812 Numerator[tU]   (* A265810 *)
%t A265812 Denominator[tU] (* A265811 *)
%t A265812 Numerator[y]    (* A265812 *)
%t A265812 Denominator[y]  (* A265813 *)
%Y A265812 Cf. A000040, A265759, A265808, A265809, A265810, A265811, A265813.
%K A265812 nonn,frac
%O A265812 1,1
%A A265812 _Clark Kimberling_, Jan 02 2016
%E A265812 More terms from _Bert Dobbelaere_, Jul 20 2022