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A265813 Denominators of primes-only best approximates (POBAs) to Pi; see Comments. 7
2, 2, 5, 7, 13, 53, 67, 71, 197, 241, 311, 353, 1427, 1667, 1723, 3023, 4591, 5113, 5749, 9817, 14563, 15241, 19309 (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 A265806 are also terms of A265801 (denominators of POBAs to tau).

LINKS

Table of n, a(n) for n=1..23.

EXAMPLE

The POBAs to Pi start with 5/2, 7/2, 17/5, 23/7, 41/13, 167/53, 211/67, 223/71, 619/197. For example, if p and q are primes and q > 53, then 167/53 is closer to Pi than p/q is.

MATHEMATICA

x = Pi; 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, A265812/A265813 *)

Numerator[tL]   (* A265808 *)

Denominator[tL] (* A265809 *)

Numerator[tU]   (* A265810 *)

Denominator[tU] (* A265811 *)

Numerator[y]    (* A265812 *)

Denominator[y]  (* A265813 *)

CROSSREFS

Cf. A000040, A265759, A265808, A265809, A265810, A265811, A265812.

Sequence in context: A095326 A095330 A036378 * A259864 A028303 A255063

Adjacent sequences:  A265810 A265811 A265812 * A265814 A265815 A265816

KEYWORD

nonn,frac,more

AUTHOR

Clark Kimberling, Jan 02 2016

STATUS

approved

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Last modified June 25 09:48 EDT 2019. Contains 324347 sequences. (Running on oeis4.)