A265765 - OEIS

%I #12 Jan 09 2016 14:23:24

%S 11,7,13,11,19,29,43,53,67,149,163,173,211,269,283,293,317,331,389,

%T 509,523,547,557,653,691,773,787,797,907,1051,1109,1123,1171,1229,

%U 1493,1531,1637,1723,1733,1867,1949,1997,2011,2083,2251,2309,2347,2371,2467

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

%C 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 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 x = 4; z = 200; 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, A265765/A120639 *)

%t Numerator[tL]   (* A162857 *)

%t Denominator[tL] (* A062737 *)

%t Numerator[tU]   (* A090866 *)

%t Denominator[tU] (* A023212 *)

%t Numerator[y]    (* A265765 *)

%t Denominator[y]  (* A120639 *)

%Y Cf. A000040, A023212, A062737, A090866, A120639, A162857.

%K nonn,frac

%O 1,1

%A _Clark Kimberling_, Dec 18 2015