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A265812 Numerators of primes-only best approximates (POBAs) to Pi; see Comments. 7
5, 7, 17, 23, 41, 167, 211, 223, 619, 757, 977, 1109, 4483, 5237, 5413, 9497, 14423, 16063, 18061, 30841, 45751, 47881, 60661, 137341, 162901, 177811, 211891, 626443, 833719, 38144863, 40436969, 45230587, 93379723, 114431749, 120059441, 185091653, 347672183, 1725229397, 1736068099 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences. Many terms of A265806 are also terms of A265801 (denominators of POBAs to tau).
LINKS
EXAMPLE
The POBAs to Pi start with 5/2, 7/2, 17/5, 23/7, 41/13, 167/53, 211/67, 223/71, 619/197. For example, if p and q are primes and q > 53, then 167/53 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: A214345 A166109 A157755 * A128352 A025085 A175238
KEYWORD
nonn,frac,more
AUTHOR
Clark Kimberling, Jan 02 2016
EXTENSIONS
More terms from Bert Dobbelaere, Jul 20 2022
STATUS
approved

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Last modified April 15 13:47 EDT 2024. Contains 371689 sequences. (Running on oeis4.)