A265788 Numerators of primes-only best approximates (POBAs) to sqrt(5); see Comments.

3, 5, 7, 11, 29, 163, 199, 521, 3571, 26683, 111667, 150427, 154841

Offset: 1

Author

Clark Kimberling, Dec 26 2015

Keywords

• nonn
• frac
• more

Comments

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

Examples

			The POBAs to sqrt(5) start with 3/2, 5/2, 7/3, 11/5, 29/13, 163/73, 199/89, 521/233. For example, if p and q are primes and q > 89, then 199/89 is closer to sqrt(5) than p/q is.
		

Crossrefs

Cf. A000040, A265759, A265784, A265785, A265786, A265788, A265789.

Programs

Mathematica

x = Sqrt[5]; 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, A265782/A265783 *)
Numerator[tL]   (* A265784 *)
Denominator[tL] (* A265785 *)
Numerator[tU]   (* A265786 *)
Denominator[tU] (* A265787 *)
Numerator[y]    (* A265788 *)
Denominator[y]  (* A265789 *)

Extensions

a(10)-a(13) from Robert Price, Apr 05 2019