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A265783 Denominators of primes-only best approximates (POBAs) to sqrt(3); see Comments. 14
2, 2, 3, 11, 41, 347, 907, 1489, 2131, 32801, 64613 (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

Table of n, a(n) for n=1..11.

EXAMPLE

The POBAs to sqrt(3) start with 5/2, 3/2, 5/3, 19/11, 71/41, 601/347, 1571/907. For example, if p and q are primes and q > 347, then 601/347 is closer to sqrt(3) than p/q is.

MATHEMATICA

x = Sqrt[3]; 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]   (* A265778 *)

Denominator[tL] (* A265779 *)

Numerator[tU]   (* A265780 *)

Denominator[tU] (* A265781 *)

Numerator[y]    (* A265782 *)

Denominator[y]  (* A265783 *)

CROSSREFS

Cf. A000040, A265759, A265778, A265779, A265780, A265781, A265782.

Sequence in context: A143931 A143933 A284708 * A246670 A075095 A178343

Adjacent sequences:  A265780 A265781 A265782 * A265784 A265785 A265786

KEYWORD

nonn,frac,more

AUTHOR

Clark Kimberling, Dec 23 2015

EXTENSIONS

a(10)-a(11) from Robert Price, Apr 05 2019

STATUS

approved

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Last modified October 19 11:26 EDT 2019. Contains 328216 sequences. (Running on oeis4.)