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A265815 Denominators of lower primes-only best approximates (POBAs) to e; see Comments. 7
2, 5, 7, 113, 163, 227, 823, 887, 1093, 2179, 2591, 2797, 4373, 4657, 5651, 8867, 27673, 32749, 47189, 104459, 155723, 430061, 583853, 673297, 1126523, 1869173, 3120317, 3445919, 8341961, 24681191, 26349383, 70271051, 77869361, 81514259, 89910487, 157461181, 533931763, 583892083, 770930497 (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 e; start with 5/2, 13/5, 19/7, 307/113, 443/163, 617/227, 2237/823. For example, if p and q are primes and q > 823, and p/q < e, then 2237/823 is closer to e than p/q is.
MATHEMATICA
x = E; 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, A265818/A265819 *)
Numerator[tL] (* A265814 *)
Denominator[tL] (* A265815 *)
Numerator[tU] (* A265816 *)
Denominator[tU] (* A265817 *)
Numerator[y] (* A265818 *)
Denominator[y] (* A265819 *)
CROSSREFS
Sequence in context: A343833 A103056 A041445 * A041961 A242169 A058854
KEYWORD
nonn,frac
AUTHOR
Clark Kimberling, Jan 02 2016
EXTENSIONS
More terms from Bert Dobbelaere, Jul 21 2022
STATUS
approved

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Last modified June 16 13:05 EDT 2024. Contains 373429 sequences. (Running on oeis4.)