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%I #15 Apr 07 2019 10:25:43
%S 5,5,31,47,157,911,1021,1487,4283,5147,8629,11069,15017,20939,22447,
%T 24709,38239,80803
%N Numerators of upper primes-only best approximates (POBAs) to the golden ratio, tau (A001622); 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 tau start with 5/2, 5/3, 31/19, 47/29, 157/97, 911/563, 1021/631. For example, if p and q are primes and q > 97, and p/q > tau, then 157/97 is closer to tau than p/q is.
%t x = GoldenRatio; 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, A265800/A265801 *)
%t Numerator[tL] (* A265796 *)
%t Denominator[tL] (* A265797 *)
%t Numerator[tU] (* A265798 *)
%t Denominator[tU] (* A265799 *)
%t Numerator[y] (* A265800 *)
%t Denominator[y] (* A265801 *)
%Y Cf. A000040, A001622, A265759, A265796, A265797, A265799, A265800, A265801.
%K nonn,frac,more
%O 1,1
%A _Clark Kimberling_, Dec 29 2015
%E a(13)-a(18) from _Robert Price_, Apr 06 2019
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