This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%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