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A265798 Numerators of upper primes-only best approximates (POBAs) to the golden ratio, tau (A001622); see Comments. 9
5, 5, 31, 47, 157, 911, 1021, 1487, 4283, 5147, 8629, 11069, 15017, 20939, 22447, 24709, 38239, 80803 (list; graph; refs; listen; history; text; internal format)
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: A117858 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

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Last modified August 17 11:51 EDT 2019. Contains 326057 sequences. (Running on oeis4.)