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%I #9 Jul 20 2022 17:17:47
%S 2,5,7,11,13,17,31,61,71,241,311,1427,1667,3023,4591,5749,9817,14563,
%T 15241,19309,43717,51853,56599,170701,177481,183809,184487,193979,
%U 194431,265381,13800151,14397343,33239959,35429437,38216107,58916503,261541507,414604999,549157573
%N Denominators of lower primes-only best approximates (POBAs) to Pi; see Comments.
%C 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 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 For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.
%e 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 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, A265810, A265811, A265812, A265813.
%K nonn,frac
%O 1,1
%A _Clark Kimberling_, Jan 02 2016
%E More terms from _Bert Dobbelaere_, Jul 20 2022
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