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A265809 Denominators of lower primes-only best approximates (POBAs) to Pi; see Comments. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
LINKS
EXAMPLE
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.
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
Sequence in context: A323782 A020591 A197188 * A303112 A182320 A119993
KEYWORD
nonn,frac
AUTHOR
Clark Kimberling, Jan 02 2016
EXTENSIONS
More terms from Bert Dobbelaere, Jul 20 2022
STATUS
approved

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Last modified July 12 17:26 EDT 2024. Contains 374251 sequences. (Running on oeis4.)