A265819 Denominators of primes-only best approximates (POBAs) to e; see Comments.

2, 2, 5, 7, 71, 2591, 2797, 4139, 5573, 5651, 6361, 8867, 17159, 18089, 1126523, 1869173, 3120317, 3245939, 8341961, 23302423, 24681191, 26349383, 52372163, 81514259, 89910487, 157461181, 475992263, 583892083, 770930497, 1184975581, 1436753887, 1498302107, 5389332217

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 POBA's. 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).

Links

Table of n, a(n) for n=1..33.

Example

The POBAs to Pi start with 7/2, 5/2, 13/5, 19/7, 193/71, 7043/2591, 7603/2797. For example, if p and q are primes and q > 2591, then 7043/2591 is closer to e than p/q is.

Mathematica

x = E; 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, A265818/A265819 *)
Numerator[tL]   (* A265814 *)
Denominator[tL] (* A265815 *)
Numerator[tU]   (* A265816 *)
Denominator[tU] (* A265817 *)
Numerator[y]    (* A265818 *)
Denominator[y]  (* A265819 *)

Crossrefs

Cf. A000040, A265759, A265814, A265815, A265816, A265817, A265818.

Sequence in context: A221835 A074476 A224610 * A254746 A011021 A077232

Adjacent sequences: A265816 A265817 A265818 * A265820 A265821 A265822

Keyword

nonn,frac,more

Author

Clark Kimberling, Jan 06 2016

Extensions

More terms from Bert Dobbelaere, Jul 21 2022

Status

approved