A265798 Numerators of upper primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments.

5, 5, 31, 47, 157, 911, 1021, 1487, 4283, 5147, 8629, 11069, 15017, 20939, 22447, 24709, 38239, 80803

Offset

1,1

Comments

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.

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).

For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.

Links

Table of n, a(n) for n=1..18.

Example

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.

Mathematica

x = GoldenRatio; z = 1000; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265800/A265801 *)
Numerator[tL]   (* A265796 *)
Denominator[tL] (* A265797 *)
Numerator[tU]   (* A265798 *)
Denominator[tU] (* A265799 *)
Numerator[y]    (* A265800 *)
Denominator[y]  (* A265801 *)

Crossrefs

Cf. A000040, A001622, A265759, A265796, A265797, A265799, A265800, A265801.

Sequence in context: A365824 A014434 A106830 * A259264 A259265 A270221

Adjacent sequences: A265795 A265796 A265797 * A265799 A265800 A265801

Keyword

nonn,frac,more

Author

Clark Kimberling, Dec 29 2015

Extensions

a(13)-a(18) from Robert Price, Apr 06 2019

Status

approved