A265812 Numerators of primes-only best approximates (POBAs) to Pi; see Comments.

5, 7, 17, 23, 41, 167, 211, 223, 619, 757, 977, 1109, 4483, 5237, 5413, 9497, 14423, 16063, 18061, 30841, 45751, 47881, 60661, 137341, 162901, 177811, 211891, 626443, 833719, 38144863, 40436969, 45230587, 93379723, 114431749, 120059441, 185091653, 347672183, 1725229397, 1736068099

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..39.

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, A265813.

Sequence in context: A214345 A166109 A157755 * A128352 A025085 A175238

Adjacent sequences: A265809 A265810 A265811 * A265813 A265814 A265815

Keyword

nonn,frac,more

Author

Clark Kimberling, Jan 02 2016

Extensions

More terms from Bert Dobbelaere, Jul 20 2022

Status

approved