A265811 Denominators of upper primes-only best approximates (POBAs) to Pi; see Comments.

2, 5, 7, 13, 53, 67, 137, 179, 181, 197, 353, 1723, 3319, 5113, 6469, 9181, 15269, 17981, 22727, 24083, 31541, 34253, 37643, 46457, 64763, 67447, 199403, 531101, 1791689, 5175551, 6369709, 12141887, 12871487, 23089051, 29723689, 36424757, 43324889, 84725681, 105426077, 110667493

Offset

1,1

Comments

Suppose that x > 0. A fraction p/q of primes is an upper primes-only best approximate, and we write "p/q is in U(x)", if p'/q < x < p/q < u/v for all primes u and v such that v < q, where p' is greatest prime < p in case p >= 3.

Let q(1) = 2 and let p(1) be the least prime >= x. The sequence U(x) follows inductively: for n >= 1, let q(n) is the least prime q such that x < p/q < p(n)/q(n) for some prime p. Let q(n+1) = q and let p(n+1) be the least prime p such that x < p/q < p(n)/q(n).

For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.

Links

Table of n, a(n) for n=1..40.

Example

The upper POBAs to Pi start with 7/2, 17/5, 23/7, 41/13, 167/53, 211/67, 431/137. For example, if p and q are primes and q > 67, and p/q > Pi, then 211/67 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, A265812, A265813.

Sequence in context: A023242 A164570 A348671 * A265793 A126338 A305993

Adjacent sequences: A265808 A265809 A265810 * A265812 A265813 A265814

Keyword

nonn,frac

Author

Clark Kimberling, Jan 02 2016

Extensions

More terms from Bert Dobbelaere, Jul 20 2022

Status

approved