cp's OEIS Frontend

A265818 Numerators of primes-only best approximates (POBAs) to e; see Comments.

%I A265818 #16 Jul 21 2022 01:53:43
%S A265818 7,5,13,19,193,7043,7603,11251,15149,15361,17291,24103,46643,49171,
%T A265818 3062207,5080939,8481901,8823377,22675801,63342553,67090433,71625049,
%U A265818 142362299,221578729,244402043,428023867,1293881119,1587183239,2095606361,3221097589,3905501983,4072807391,14649723833
%N A265818 Numerators of primes-only best approximates (POBAs) to e; see Comments.
%C A265818 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 A265818 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 A265818 x = E; z = 1000; p[k_] := p[k] = Prime[k];
%t A265818 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265818 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265818 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265818 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265818 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265818 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265818 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265818/A265819 *)
%t A265818 Numerator[tL]   (* A265814 *)
%t A265818 Denominator[tL] (* A265815 *)
%t A265818 Numerator[tU]   (* A265816 *)
%t A265818 Denominator[tU] (* A265817 *)
%t A265818 Numerator[y]    (* A265818 *)
%t A265818 Denominator[y]  (* A265819 *)
%Y A265818 Cf. A000040, A265759, A265814, A265815, A265816, A265817, A265819.
%K A265818 nonn,frac
%O A265818 1,1
%A A265818 _Clark Kimberling_, Jan 06 2016
%E A265818 More terms from _Bert Dobbelaere_, Jul 21 2022