A265763 Numerators of primes-only best approximates (POBAs) to 3; see Comments.

7, 5, 17, 13, 23, 19, 31, 41, 37, 53, 59, 71, 67, 89, 113, 109, 131, 127, 139, 157, 179, 181, 199, 211, 239, 251, 269, 293, 311, 307, 337, 383, 379, 409, 419, 449, 491, 487, 503, 499, 521, 541, 571, 577, 593, 599, 631, 683, 701, 719, 751, 773, 769, 787, 809

Offset: 1

Author

Clark Kimberling, Dec 18 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 for 3 start with 7/2, 5/2, 17/5, 13/5, 23/7, 19/7, 31/11, 41/13, 37/13, 53/17. For example, if p and q are primes and q > 13, then 41/13 is closer to 3 than p/q is.
		

Crossrefs

Cf. A000040, A023208, A088878, A091180, A094525, A222565.

Programs

Mathematica

x = 3; 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, A265763/A265764 *)
Numerator[tL]   (* A091180 *)
Denominator[tL] (* A088878 *)
Numerator[tU]   (* A094525 *)
Denominator[tU] (* A023208 *)
Numerator[y]    (* A265763 *)
Denominator[y]  (* A265764 *)