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

Author

Clark Kimberling, Jan 02 2016

Keywords

• nonn
• frac

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).

Examples

			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.
		

Crossrefs

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