A265813 Denominators of primes-only best approximates (POBAs) to Pi; see Comments.

2, 2, 5, 7, 13, 53, 67, 71, 197, 241, 311, 353, 1427, 1667, 1723, 3023, 4591, 5113, 5749, 9817, 14563, 15241, 19309, 43717, 51853, 56599, 67447, 199403, 265381, 12141887, 12871487, 14397343, 29723689, 36424757, 38216107, 58916503, 110667493, 549157573, 552607639

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, 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

Extensions

More terms from Bert Dobbelaere, Jul 20 2022

Status

approved