cp's OEIS Frontend

A265816 Numerators of upper primes-only best approximates (POBAs) to e; see Comments.

%I A265816 #10 Jul 21 2022 01:53:36
%S A265816 7,17,23,31,47,79,193,11251,15149,17291,25261,46643,49171,6105367,
%T A265816 8522909,8823377,42983231,63342553,97109039,97947667,142362299,
%U A265816 292315979,361821233,456318767,677946667,707276879,1161377509,1293881119,2001108827,3221097589,4154291129,7294989463,14281444873
%N A265816 Numerators of upper primes-only best approximates (POBAs) to e; see Comments.
%C A265816 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 A265816 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 A265816 For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.
%e A265816 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 A265816 x = E; z = 1000; p[k_] := p[k] = Prime[k];
%t A265816 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265816 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265816 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265816 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265816 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265816 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265816 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265818/A265819 *)
%t A265816 Numerator[tL]   (* A265814 *)
%t A265816 Denominator[tL] (* A265815 *)
%t A265816 Numerator[tU]   (* A265816 *)
%t A265816 Denominator[tU] (* A265817 *)
%t A265816 Numerator[y]    (* A265818 *)
%t A265816 Denominator[y]  (* A265819 *)
%Y A265816 Cf. A000040, A265759, A265814, A265815, A265817, A265818, A265819.
%K A265816 nonn,frac
%O A265816 1,1
%A A265816 _Clark Kimberling_, Jan 02 2016
%E A265816 More terms from _Bert Dobbelaere_, Jul 21 2022