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A265808 Numerators of lower primes-only best approximates (POBAs) to Pi; see Comments. 7
5, 13, 19, 31, 37, 53, 97, 191, 223, 757, 977, 4483, 5237, 9497, 14423, 18061, 30841, 45751, 47881, 60661 (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

Table of n, a(n) for n=1..20.

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, A265809, A265810, A265811, A265812, A265813.

Sequence in context: A129919 A045454 A166575 * A265790 A002540 A290515

Adjacent sequences:  A265805 A265806 A265807 * A265809 A265810 A265811

KEYWORD

nonn,frac,more

AUTHOR

Clark Kimberling, Jan 02 2016

STATUS

approved

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Last modified July 20 01:38 EDT 2019. Contains 325168 sequences. (Running on oeis4.)