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A265807 Denominators of primes-only best approximates (POBAs) to 1/(golden ratio) = 1/tau; see Comments. 3
2, 3, 5, 31, 37, 47, 157, 571, 911, 1021, 1487, 2351, 3571, 24709, 25463, 69247, 80803 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences. Many terms of A265807 are also terms of A265800 (numerators of POBAs to tau).

LINKS

Table of n, a(n) for n=1..17.

EXAMPLE

The POBAs to 1/tau start with 2/2, 2/3, 3/5, 19/31, 23/37, 29/47, 97/157, 353/571. For example, if p and q are primes and q > 157, then 97/157 is closer to 1/tau than p/q is.

MATHEMATICA

x = 1/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, A265806/A265807 *)

Numerator[tL] (* A265799 *)

Denominator[tL] (* A265798 *)

Numerator[tU] (* A265797 *)

Denominator[tU] (* A265796 *)

Numerator[y] (* A265806 *)

Denominator[y] (* A265807 *)

CROSSREFS

Cf. A000040, A265759, A265799, A265798, A265797, A265796, A265806.

Sequence in context: A060301 A040119 A186635 * A106308 A036797 A163079

Adjacent sequences: A265804 A265805 A265806 * A265808 A265809 A265810

KEYWORD

nonn,frac,more

AUTHOR

Clark Kimberling, Jan 02 2016

EXTENSIONS

a(14)-a(17) from Robert Price, Apr 06 2019

STATUS

approved

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Last modified February 6 18:02 EST 2023. Contains 360111 sequences. (Running on oeis4.)