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A265810 Numerators of upper primes-only best approximates (POBAs) to Pi; see Comments. 7
7, 17, 23, 41, 167, 211, 431, 563, 569, 619, 1109, 5413, 10427, 16063, 20323, 28843, 47969, 56489, 71399, 75659 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Suppose that x > 0. A fraction p/q of primes is an upper primes-only best approximate, and we write "p/q is in U(x)", if p'/q < x < p/q < u/v for all primes u and v such that v < q, where p' is greatest prime < p in case p >= 3.

Let q(1) = 2 and let p(1) be the least prime >= x. The sequence U(x) follows inductively: for n >= 1, let q(n) is the least prime q such that x < p/q < p(n)/q(n) for some prime p. Let q(n+1) = q and let p(n+1) be the least prime p such that x < p/q < p(n)/q(n).

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

LINKS

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

EXAMPLE

The upper POBAs to Pi start with 7/2, 17/5, 23/7, 41/13, 167/53, 211/67, 431/137. For example, if p and q are primes and q > 67, and p/q > Pi, then 211/67 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, A265809, A265811, A265812, A265813.

Sequence in context: A265792 A322669 A115591 * A026349 A057183 A076293

Adjacent sequences:  A265807 A265808 A265809 * A265811 A265812 A265813

KEYWORD

nonn,frac,more

AUTHOR

Clark Kimberling, Jan 02 2016

STATUS

approved

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Last modified July 23 19:33 EDT 2019. Contains 325263 sequences. (Running on oeis4.)