A265776 Numerators of primes-only best approximates (POBAs) to sqrt(2); see Comments.

2, 3, 7, 41, 977, 1093, 1373, 1427, 3701, 8597, 22247, 38287, 53569, 61927, 78643

Offset

1,1

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.

Links

Table of n, a(n) for n=1..15.

Example

The POBAs to sqrt(2) start with 2/2, 3/2, 7/5, 41/29, 977/691, 1093/773, 1373/971, 1427/1009. For example, if p and q are primes and q > 29, then 41/29 is closer to sqrt(2) than p/q is.

Mathematica

x = Sqrt[2]; z = 800; 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, A265776/A265777 *)
Numerator[tL]   (* A265772 *)
Denominator[tL] (* A265773 *)
Numerator[tU]   (* A265774 *)
Denominator[tU] (* A265775 *)
Numerator[y]    (* A265776 *)
Denominator[y]  (* A265777 *)

Crossrefs

Cf. A000040, A265759, A265772, A265773, A265774, A265775, A265777.

Sequence in context: A355129 A102604 A119662 * A163157 A260819 A000945

Adjacent sequences: A265773 A265774 A265775 * A265777 A265778 A265779

Keyword

nonn,frac,more

Author

Clark Kimberling, Dec 20 2015

Extensions

a(11)-a(15) from Robert Price, Apr 05 2019

Status

approved