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A265766 Numerators of lower primes-only best approximates (POBAs) to 5; see Comments. 5
7, 13, 23, 53, 83, 113, 233, 263, 293, 353, 443, 503, 563, 653, 683, 743, 863, 953, 983, 1163, 1193, 1283, 1553, 1583, 1733, 1913, 2003, 2153, 2213, 2243, 2333, 2393, 2543, 2843, 2963, 3083, 3203, 3413, 3593, 3803, 3863, 4133, 4283, 4643, 4703, 4733, 5153 (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 5 start with 7/2, 13/3, 23/5, 53/11, 83/17, 113/23, 233/47. For example, if p and q are primes and q > 17, and p/q < 5, then 83/17 is closer to 5 than p/q is.
MATHEMATICA
x = 5; z = 200; 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, A265768/A265769 *)
Numerator[tL] (* A265766 *)
Denominator[tL] (* A158318 *)
Numerator[tU] (* A265767 *)
Denominator[tU] (* A023217 *)
Numerator[y] (* A222568 *)
Denominator[y] (* A265769 *)
CROSSREFS
Sequence in context: A304671 A075642 A316118 * A228727 A286916 A032690
KEYWORD
nonn,frac
AUTHOR
Clark Kimberling, Dec 19 2015
STATUS
approved

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Last modified July 22 15:23 EDT 2024. Contains 374513 sequences. (Running on oeis4.)