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A265819 Denominators of primes-only best approximates (POBAs) to e; see Comments.

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%I A265819 #15 Jul 21 2022 01:53:39
%S A265819 2,2,5,7,71,2591,2797,4139,5573,5651,6361,8867,17159,18089,1126523,
%T A265819 1869173,3120317,3245939,8341961,23302423,24681191,26349383,52372163,
%U A265819 81514259,89910487,157461181,475992263,583892083,770930497,1184975581,1436753887,1498302107,5389332217
%N A265819 Denominators of primes-only best approximates (POBAs) to e; see Comments.
%C A265819 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 POBA's.  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 A265819 The POBAs to Pi start with 7/2, 5/2, 13/5, 19/7, 193/71, 7043/2591, 7603/2797. For example, if p and q are primes and q > 2591, then 7043/2591 is closer to e than p/q is.
%t A265819 x = E; z = 1000; p[k_] := p[k] = Prime[k];
%t A265819 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265819 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265819 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265819 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265819 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265819 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265819 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265818/A265819 *)
%t A265819 Numerator[tL]   (* A265814 *)
%t A265819 Denominator[tL] (* A265815 *)
%t A265819 Numerator[tU]   (* A265816 *)
%t A265819 Denominator[tU] (* A265817 *)
%t A265819 Numerator[y]    (* A265818 *)
%t A265819 Denominator[y]  (* A265819 *)
%Y A265819 Cf. A000040, A265759, A265814, A265815, A265816, A265817, A265818.
%K A265819 nonn,frac,more
%O A265819 1,1
%A A265819 _Clark Kimberling_, Jan 06 2016
%E A265819 More terms from _Bert Dobbelaere_, Jul 21 2022