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%I #10 Jul 21 2022 01:53:36
%S 7,17,23,31,47,79,193,11251,15149,17291,25261,46643,49171,6105367,
%T 8522909,8823377,42983231,63342553,97109039,97947667,142362299,
%U 292315979,361821233,456318767,677946667,707276879,1161377509,1293881119,2001108827,3221097589,4154291129,7294989463,14281444873
%N Numerators of upper primes-only best approximates (POBAs) to e; 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 e start with 77/2, 17/5, 23/7, 31/11, 47/17, 79/29, 193/71, 11251/4139. For example, if p and q are primes and q > 71, and p/q > e, then 193/71 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, A265814, A265815, 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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