A265770 Numerators of primes-only best approximates (POBAs) to 6; see Comments.

13, 11, 19, 17, 31, 29, 43, 41, 67, 79, 103, 101, 113, 139, 137, 173, 223, 257, 283, 281, 317, 353, 367, 401, 439, 499, 607, 619, 617, 643, 641, 653, 677, 761, 787, 823, 821, 907, 941, 977, 1039, 1087, 1181, 1193, 1361, 1373, 1399, 1433, 1447, 1543, 1579

Offset: 1

Author

Clark Kimberling, Dec 20 2015

Keywords

• nonn
• frac

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 6 start with 13/2, 11/2, 19/3, 17/3, 31/5, 29/5, 43/7, 41/7, 67/11, 79/13, 103/17, 101/17. For example, if p and q are primes and q > 17, then 103/17 (and 101/17) is closer to 6 than p/q is.
		

Crossrefs

Cf. A000040, A265759, A227756, A158015, A051644, A007693, A265771.

Programs

Mathematica

x = 6; 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, A265770/A265771 *)
Numerator[tL]   (* A227756 *)
Denominator[tL] (* A158015 *)
Numerator[tU]   (* A051644 *)
Denominator[tU] (* A007693 *)
Numerator[y]    (* A222570 *)
Denominator[y]  (* A265771 *)