cp's OEIS Frontend

A265808 Numerators of lower primes-only best approximates (POBAs) to Pi; see Comments.

%I A265808 #10 Jul 20 2022 17:17:57
%S A265808 5,13,19,31,37,53,97,191,223,757,977,4483,5237,9497,14423,18061,30841,
%T A265808 45751,47881,60661,137341,162901,177811,536273,557573,577453,579583,
%U A265808 609403,610823,833719,43354453,45230587,104426411,111304859,120059441,185091653,821656877,1302520019
%N A265808 Numerators of lower primes-only best approximates (POBAs) to Pi; see Comments.
%C A265808 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 A265808 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 A265808 For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.
%e A265808 The lower POBAs to Pi start with 5/2, 13/5, 19/7, 31/11, 37/13, 53/17, 97/31, 191/61, 223/71, 757/241, 977/311. For example, if p and q are primes and q > 241, and p/q < Pi, then 757/241 is closer to Pi than p/q is.
%t A265808 x = Pi; z = 1000; p[k_] := p[k] = Prime[k];
%t A265808 t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265808 d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t A265808 t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t A265808 d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t A265808 v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t A265808 b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t A265808 y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265812/A265813 *)
%t A265808 Numerator[tL]   (* A265808 *)
%t A265808 Denominator[tL] (* A265809 *)
%t A265808 Numerator[tU]   (* A265810 *)
%t A265808 Denominator[tU] (* A265811 *)
%t A265808 Numerator[y]    (* A265812 *)
%t A265808 Denominator[y]  (* A265813 *)
%Y A265808 Cf. A000040, A265759, A265809, A265810, A265811, A265812, A265813.
%K A265808 nonn,frac
%O A265808 1,1
%A A265808 _Clark Kimberling_, Jan 02 2016
%E A265808 More terms from _Bert Dobbelaere_, Jul 20 2022