A265809 Denominators of lower primes-only best approximates (POBAs) to Pi; see Comments.

2, 5, 7, 11, 13, 17, 31, 61, 71, 241, 311, 1427, 1667, 3023, 4591, 5749, 9817, 14563, 15241, 19309, 43717, 51853, 56599, 170701, 177481, 183809, 184487, 193979, 194431, 265381, 13800151, 14397343, 33239959, 35429437, 38216107, 58916503, 261541507, 414604999, 549157573

Offset: 1

Author

Clark Kimberling, Jan 02 2016

Keywords

• nonn
• frac

Comments

Suppose that x > 0. A fraction p/q of primes is a lower primes-only best approximate, and we write "p/q is in L(x)", if u/v < p/q < x < p'/q for all primes u and v such that v < q, where p' is least prime > p.

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

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

Examples

			The lower POBAs to Pi start with 5/2, 13/5, 19/7, 31/11, 37/13, 53/17, 97/31, 191/61, 223/71, 757/241, 977/311. For example, if p and q are primes and q > 241, and p/q < Pi, then 757/241 is closer to Pi than p/q is.
		

Crossrefs

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