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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, 99089, 107609, 118259, 145949, 203459, 211891, 626443, 1668503, 5628757, 16259473, 20011031, 38144863, 40436969, 72536393, 93379723, 114431749, 136109153, 266173577
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.
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 *)
KEYWORD
nonn,frac
AUTHOR
Clark Kimberling, Jan 02 2016
EXTENSIONS
More terms from Bert Dobbelaere, Jul 20 2022
STATUS
approved