login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A265789 Denominators of primes-only best approximates (POBAs) to sqrt(5); see Comments. 7
2, 2, 3, 5, 13, 73, 89, 233, 1597, 11933, 49939, 67273, 69247 (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.
LINKS
EXAMPLE
The POBAs to sqrt(5) start with 3/2, 5/2, 7/3, 11/5, 29/13, 163/73, 199/89, 521/233. For example, if p and q are primes and q > 89, then 199/89 is closer to sqrt(5) than p/q is.
MATHEMATICA
x = Sqrt[5]; 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, A265782/A265783 *)
Numerator[tL] (* A265784 *)
Denominator[tL] (* A265785 *)
Numerator[tU] (* A265786 *)
Denominator[tU] (* A265787 *)
Numerator[y] (* A265788 *)
Denominator[y] (* A265789 *)
CROSSREFS
Sequence in context: A111264 A065994 A152446 * A055937 A090255 A228162
KEYWORD
nonn,frac,more
AUTHOR
Clark Kimberling, Dec 29 2015
EXTENSIONS
a(10)-a(13) from Robert Price, Apr 05 2019
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)