A265806 Numerators of primes-only best approximates (POBAs) to 1/(golden ratio) = 1/tau; see Comments.

2, 2, 3, 19, 23, 29, 97, 353, 563, 631, 919, 1453, 2207, 15271, 15737, 42797, 49939

Offset: 1

Author

Clark Kimberling, Jan 02 2016

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. Many terms of A265806 are also terms of A265801 (denominators of POBAs to tau).

Examples

			The POBAs to 1/tau start with 2/2, 2/3, 3/5, 19/31, 23/37, 29/47, 97/157, 353/571. For example, if p and q are primes and q > 157, then 97/157 is closer to 1/tau than p/q is.
		

Crossrefs

Cf. A000040, A265759, A265799, A265798, A265797, A265796, A265807.

Programs

Mathematica

x = 1/GoldenRatio; 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, A265806/A265807 *)
Numerator[tL]   (* A265799 *)
Denominator[tL] (* A265798 *)
Numerator[tU]   (* A265797 *)
Denominator[tU] (* A265796 *)
Numerator[y]    (* A265806 *)
Denominator[y]  (* A265807 *)

Extensions

a(14)-a(17) from Robert Price, Apr 06 2019