cp's OEIS Frontend

A265790 Numerators of lower primes-only best approximates (POBAs) to sqrt(8); see Comments.

%I A265790 #10 Apr 06 2019 10:53:58
%S A265790 5,13,19,31,223,359,461,659,6163,8539,8737,25453,32377,35839
%N A265790 Numerators of lower primes-only best approximates (POBAs) to sqrt(8); see Comments.
%C A265790 Suppose that x > 0. A fraction p/q of primes is a lower primes-only best approximate, and we write "p/q is in L(x)", if u/v < p/q < x < p'/q for all primes u and v such that v < q, where p' is least prime > p.
%C A265790 Let q(1) be the least prime q such that u/q < x for some prime u, and let p(1) be the greatest such u. The sequence L(x) follows inductively: for n > 1, let q(n) is the least prime q such that p(n)/q(n) < p/q < x for some prime p. Let q(n+1) = q and let p(n+1) be the greatest prime p such that p(n)/q(n) < p/q < x.
%C A265790 For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.
%e A265790 The lower POBAs to sqrt(8) start with 5/2, 13/5, 19/7, 31/11, 223/79, 359/127. For example, if p and q are primes and q > 79, and p/q < sqrt(8), then 223/79 is closer to sqrt(8) than p/q is.
%t A265790 x = Sqrt[8]; z = 1000; p[k_] := p[k] = Prime[k];
%t A265790 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265790 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265790 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265790 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265790 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265790 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265790 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265782/A265783 *)
%t A265790 Numerator[tL]   (* A265790 *)
%t A265790 Denominator[tL] (* A265791 *)
%t A265790 Numerator[tU]   (* A265792 *)
%t A265790 Denominator[tU] (* A265793 *)
%t A265790 Numerator[y]    (* A265794 *)
%t A265790 Denominator[y]  (* A265795 *)
%Y A265790 Cf. A000040, A265759, A265791, A265792, A265793, A265794, A265795.
%K A265790 nonn,frac,more
%O A265790 1,1
%A A265790 _Clark Kimberling_, Dec 29 2015
%E A265790 a(12)-a(14) from _Robert Price_, Apr 06 2019