

A265801


Denominators of primesonly best approximates (POBAs) to the golden ratio, tau; see Comments.


12



2, 2, 3, 7, 19, 23, 29, 97, 353, 563, 631, 919, 1453, 2207, 15271, 15737, 42797, 49939
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OFFSET

1,1


COMMENTS

Suppose that x > 0. A fraction p/q of primes is a primesonly 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.
Is this related to A165571?  R. J. Mathar, Jan 10 2016


LINKS

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


EXAMPLE

The POBAs to tau start with 5/2, 3/2, 5/3, 11/7, 31/19, 37/23, 47/29, 157/97, 571/353, 911/563. For example, if p and q are primes and q > 29, then 47/29 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, A265759, A265796, A265797, A265798, A265799, A265800.
Sequence in context: A117387 A113842 A032161 * A098738 A291742 A083701
Adjacent sequences: A265798 A265799 A265800 * A265802 A265803 A265804


KEYWORD

nonn,frac,more


AUTHOR

Clark Kimberling, Jan 02 2016


EXTENSIONS

a(15)a(18) from Robert Price, Apr 06 2019


STATUS

approved



