cp's OEIS Frontend

A265765 Numerators of primes-only best approximates (POBAs) to 4; see Comments.

%I A265765 #12 Jan 09 2016 14:23:24
%S A265765 11,7,13,11,19,29,43,53,67,149,163,173,211,269,283,293,317,331,389,
%T A265765 509,523,547,557,653,691,773,787,797,907,1051,1109,1123,1171,1229,
%U A265765 1493,1531,1637,1723,1733,1867,1949,1997,2011,2083,2251,2309,2347,2371,2467
%N A265765 Numerators of primes-only best approximates (POBAs) to 4; see Comments.
%C A265765 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 A265765 The POBAs for 4 start with  11/2, 7/2, 13/3, 11/3, 19/5, 29/7, 43/11, 53/13, 67/17. For example, if p and q are primes and q > 13, then 53/13 is closer to 3 than p/q is.
%t A265765 x = 4; z = 200; p[k_] := p[k] = Prime[k];
%t A265765 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265765 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265765 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265765 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265765 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265765 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265765 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265765/A120639 *)
%t A265765 Numerator[tL]   (* A162857 *)
%t A265765 Denominator[tL] (* A062737 *)
%t A265765 Numerator[tU]   (* A090866 *)
%t A265765 Denominator[tU] (* A023212 *)
%t A265765 Numerator[y]    (* A265765 *)
%t A265765 Denominator[y]  (* A120639 *)
%Y A265765 Cf. A000040, A023212, A062737, A090866, A120639, A162857.
%K A265765 nonn,frac
%O A265765 1,1
%A A265765 _Clark Kimberling_, Dec 18 2015