|
%I #10 Jul 20 2022 17:09:04
%S 2,5,7,13,53,67,137,179,181,197,353,1723,3319,5113,6469,9181,15269,
%T 17981,22727,24083,31541,34253,37643,46457,64763,67447,199403,531101,
%U 1791689,5175551,6369709,12141887,12871487,23089051,29723689,36424757,43324889,84725681,105426077,110667493
%N Denominators of upper primes-only best approximates (POBAs) to Pi; see Comments.
%C 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 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 For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.
%e The upper POBAs to Pi start with 7/2, 17/5, 23/7, 41/13, 167/53, 211/67, 431/137. For example, if p and q are primes and q > 67, and p/q > Pi, then 211/67 is closer to Pi than p/q is.
%t x = Pi; z = 1000; p[k_] := p[k] = Prime[k];
%t t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
%t d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
%t t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
%t d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
%t v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
%t b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
%t y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265812/A265813 *)
%t Numerator[tL] (* A265808 *)
%t Denominator[tL] (* A265809 *)
%t Numerator[tU] (* A265810 *)
%t Denominator[tU] (* A265811 *)
%t Numerator[y] (* A265812 *)
%t Denominator[y] (* A265813 *)
%Y Cf. A000040, A265759, A265808, A265809, A265810, A265812, A265813.
%K nonn,frac
%O 1,1
%A _Clark Kimberling_, Jan 02 2016
%E More terms from _Bert Dobbelaere_, Jul 20 2022
|
