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%I #10 Jul 21 2022 01:53:51
%S 5,13,19,307,443,617,2237,2411,2971,5923,7043,7603,11887,12659,15361,
%T 24103,75223,89021,128273,283949,423299,1169027,1587077,1830211,
%U 3062207,5080939,8481901,9366979,22675801,67090433,71625049,191016521,211670869,221578729,244402043,428023867,1451377009
%N Numerators of lower primes-only best approximates (POBAs) to e; 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 e; start with 5/2, 13/5, 19/7, 307/113, 443/163, 617/227, 2237/823. For example, if p and q are primes and q > 823, and p/q < e, then 2237/823 is closer to e than p/q is.
%t x = E; 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, A265818/A265819 *)
%t Numerator[tL] (* A265814 *)
%t Denominator[tL] (* A265815 *)
%t Numerator[tU] (* A265816 *)
%t Denominator[tU] (* A265817 *)
%t Numerator[y] (* A265818 *)
%t Denominator[y] (* A265819 *)
%Y Cf. A000040, A265759, A265815, A265816, A265817, A265818, A265819.
%K nonn,frac
%O 1,1
%A _Clark Kimberling_, Jan 02 2016
%E More terms from _Bert Dobbelaere_, Jul 21 2022
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