A265764 - OEIS

%I #12 Jan 09 2016 14:23:15

%S 2,2,5,5,7,7,11,13,13,17,19,23,23,29,37,37,43,43,47,53,59,61,67,71,79,

%T 83,89,97,103,103,113,127,127,137,139,149,163,163,167,167,173,181,191,

%U 193,197,199,211,227,233,239,251,257,257,263,269,271,277,293

%N Denominators of primes-only best approximates (POBAs) to 3; 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 3 start with  7/2, 5/2, 17/5, 13/5, 23/7, 19/7, 31/11, 41/13, 37/13, 53/17. For example, if p and q are primes and q > 13, then 41/13 is closer to 3 than p/q is.

%t x = 3; 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, A265763/A265764 *)

%t Numerator[tL]   (* A091180 *)

%t Denominator[tL] (* A088878 *)

%t Numerator[tU]   (* A094525 *)

%t Denominator[tU] (* A023208 *)

%t Numerator[y]    (* A265763 *)

%t Denominator[y]  (* A265764 *)

%Y Cf. A000040, A023208, A088878, A091180, A094525, A265763.

%K nonn,frac

%O 1,1

%A _Clark Kimberling_, Dec 18 2015