This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (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 Cf. A000040, A265759, A265808, A265810, A265811, A265812, A265813. Sequence in context: A323782 A020591 A197188 * A303112 A182320 A119993 Adjacent sequences:  A265806 A265807 A265808 * A265810 A265811 A265812 KEYWORD nonn,frac,more AUTHOR Clark Kimberling, Jan 02 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 14 17:31 EDT 2019. Contains 328022 sequences. (Running on oeis4.)