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

Author

Clark Kimberling, Jan 02 2016

Keywords

• nonn
• frac

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.

Examples

			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.
		

Crossrefs

Cf. A000040, A265759, A265808, A265809, A265810, A265812, A265813.

Programs

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 *)

Extensions

More terms from Bert Dobbelaere, Jul 20 2022