A265767 Numerators of upper primes-only best approximates (POBAs) to 5; see Comments.

11, 37, 67, 97, 157, 307, 337, 367, 397, 487, 547, 757, 787, 907, 967, 997, 1117, 1567, 1657, 1747, 1867, 1987, 2287, 2437, 2617, 2707, 2857, 2887, 3037, 3067, 3217, 3307, 3457, 3547, 3637, 3697, 3847, 4057, 4297, 4597, 4957, 4987, 5107, 5167, 5197, 5347

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..46.

Example

The upper POBAs to 5 start with 11/2, 37/7, 67/13, 97/19, 157/31, 307/61, 337/67, 367/73. For example, if p and q are primes and q > 19, and p/q > 5, then 97/19 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

Cf. A000040, A023217, A158318, A265759, A265766, A222568, A265769.

Sequence in context: A116057 A099227 A122728 * A031381 A358079 A160023

Adjacent sequences: A265764 A265765 A265766 * A265768 A265769 A265770

Keyword

nonn,frac

Author

Clark Kimberling, Dec 19 2015

Status

approved