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

Author

Clark Kimberling, Dec 19 2015

Keywords

• nonn
• frac

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.

Examples

			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.
		

Crossrefs

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

Programs

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 *)