cp's OEIS Frontend

A265769 Denominators of primes-only best approximates (POBAs) to 5; see Comments.

%I A265769 #6 Dec 23 2015 14:16:07
%S A265769 2,2,5,7,11,13,17,19,23,31,47,53,59,61,67,71,73,79,89,97,101,109,113,
%T A265769 131,137,149,151,157,173,181,191,193,197,199,223,233,239,257,311,313,
%U A265769 317,331,347,349,373,383,397,401,431,443,449,457,467,479,487,509
%N A265769 Denominators of primes-only best approximates (POBAs) to 5; see Comments.
%C A265769 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 A265769 The POBAs to 5 start with 7/2, 11/2, 23/5, 37/7, 53/11, 67/13, 83/17, 97/19, 113/23, 157/31, 233/47. For example, if p and q are primes and q > 13, then 67/13 is closer to 5 than p/q is.
%t A265769 x = 5; z = 200; p[k_] := p[k] = Prime[k];
%t A265769 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265769 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265769 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265769 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265769 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265769 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265769 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265768/A265769 *)
%t A265769 Numerator[tL]   (* A265766 *)
%t A265769 Denominator[tL] (* A158318 *)
%t A265769 Numerator[tU]   (* A265767 *)
%t A265769 Denominator[tU] (* A023217 *)
%t A265769 Numerator[y]    (* A222568 *)
%t A265769 Denominator[y]  (* A265769 *)
%Y A265769 Cf. A000040, A265759, A265766, A158318, A265767, A023217, A265768.
%K A265769 nonn,frac
%O A265769 1,1
%A A265769 _Clark Kimberling_, Dec 20 2015