A265786 - OEIS

%I #10 Apr 06 2019 01:09:24

%S 5,7,43,83,293,709,937,1259,2131,6791,8951,12721,26683,111667,154841

%N Numerators of upper primes-only best approximates (POBAs) to sqrt(5); 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 sqrt(5) start with 5/2, 7/3, 43/19, 83/37, 293/131, 709/317, 937/419. For example, if p and q are primes and q > 131, and p/q > sqrt(5), then 293/131 is closer to sqrt(5) than p/q is.

%t x = Sqrt[5]; 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, A265782/A265783 *)

%t Numerator[tL]   (* A265784 *)

%t Denominator[tL] (* A265785 *)

%t Numerator[tU]   (* A265786 *)

%t Denominator[tU] (* A265787 *)

%t Numerator[y]    (* A222588 *)

%t Denominator[y]  (* A265789 *)

%Y Cf. A000040, A265759, A265784, A265785, A265787, A265788, A265789.

%K nonn,frac,more

%O 1,1

%A _Clark Kimberling_, Dec 23 2015

%E a(13)-a(15) from _Robert Price_, Apr 05 2019