cp's OEIS Frontend

A265817 Denominators of upper primes-only best approximates (POBAs) to e; see Comments.

%I A265817 #8 Jul 21 2022 01:53:47
%S A265817 2,5,7,11,17,29,71,4139,5573,6361,9293,17159,18089,2246039,3135403,
%T A265817 3245939,15812647,23302423,35724419,36032933,52372163,107537039,
%U A265817 133106593,167870293,249402641,260192623,427246909,475992263,736166797,1184975581,1528278299,2683676647,5253849959,5389332217
%N A265817 Denominators of upper primes-only best approximates (POBAs) to e; see Comments.
%C A265817 Suppose that x > 0. A fraction p/q of primes is an upper primes-only best approximate, and we write "p/q is in U(x)", if p'/q < x < p/q < u/v for all primes u and v such that v < q, where p' is greatest prime < p in case p >= 3.
%C A265817 Let q(1) = 2 and let p(1) be the least prime >= x. The sequence U(x) follows inductively:  for n >= 1, let q(n) is the least prime q such that x < p/q < p(n)/q(n) for some prime p.  Let q(n+1) = q and let p(n+1) be the least prime p such that x < p/q < p(n)/q(n).
%C A265817 For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.
%e A265817 The upper POBAs to e start with 77/2, 17/5, 23/7, 31/11, 47/17, 79/29, 193/71, 11251/4139.  For example, if p and q are primes and q > 71, and p/q > e, then 193/71 is closer to e than p/q is.
%t A265817 x = E; z = 1000; p[k_] := p[k] = Prime[k];
%t A265817 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265817 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (*lower POBA*)
%t A265817 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265817 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (*upper POBA*)
%t A265817 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265817 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265817 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (*POBA,A265818/A265819*)
%t A265817 Numerator[tL]   (*A265814*)
%t A265817 Denominator[tL] (*A265815*)
%t A265817 Numerator[tU]   (*A265816*)
%t A265817 Denominator[tU] (*A265817*)
%t A265817 Numerator[y]    (*A265818*)
%t A265817 Denominator[y]  (*A265819*)
%Y A265817 Cf. A000040, A265759, A265814, A265815, A265816, A265818, A265819.
%K A265817 nonn,frac
%O A265817 1,1
%A A265817 _Clark Kimberling_, Jan 06 2016
%E A265817 More terms from _Bert Dobbelaere_, Jul 21 2022