%I
%S 2,2,5,7,13,53,67,71,197,241,311,353,1427,1667,1723,3023,4591,5113,
%T 5749,9817,14563,15241,19309
%N Denominators of primesonly best approximates (POBAs) to Pi; see Comments.
%C Suppose that x > 0. A fraction p/q of primes is a primesonly 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 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 x = Pi; 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, A265812/A265813 *)
%t Numerator[tL] (* A265808 *)
%t Denominator[tL] (* A265809 *)
%t Numerator[tU] (* A265810 *)
%t Denominator[tU] (* A265811 *)
%t Numerator[y] (* A265812 *)
%t Denominator[y] (* A265813 *)
%Y Cf. A000040, A265759, A265808, A265809, A265810, A265811, A265812.
%K nonn,frac,more
%O 1,1
%A _Clark Kimberling_, Jan 02 2016
