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Denominator of lower primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.
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%I #12 Apr 07 2019 00:01:52

%S 2,7,23,101,107,149,353,761,971,1453,2207,15737,42797

%N Denominator of lower 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 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 tau start with 3/2, 11/7, 37/23, 163/101, 173/107, 241/149. For example, if p and q are primes and q > 101, and p/q < tau, then 163/101 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, A265798, A265799, A265800, A265801.

%K nonn,frac,more

%O 1,1

%A _Clark Kimberling_, Dec 29 2015

%E a(12)-a(13) from _Robert Price_, Apr 06 2019