A265816 Numerators of upper primes-only best approximates (POBAs) to e; see Comments.

7, 17, 23, 31, 47, 79, 193, 11251, 15149, 17291, 25261, 46643, 49171, 6105367, 8522909, 8823377, 42983231, 63342553, 97109039, 97947667, 142362299, 292315979, 361821233, 456318767, 677946667, 707276879, 1161377509, 1293881119, 2001108827, 3221097589, 4154291129, 7294989463, 14281444873

Offset

1,1

Comments

Suppose that x > 0. A fraction p/q of primes is an upper primes-only best approximate, and we write "p/q is in U(x)", if p'/q < x < p/q < u/v for all primes u and v such that v < q, where p' is greatest prime < p in case p >= 3.

Let q(1) = 2 and let p(1) be the least prime >= x. The sequence U(x) follows inductively: for n >= 1, let q(n) is the least prime q such that x < p/q < p(n)/q(n) for some prime p. Let q(n+1) = q and let p(n+1) be the least prime p such that x < p/q < p(n)/q(n).

For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.

Links

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

Example

The upper POBAs to e start with 77/2, 17/5, 23/7, 31/11, 47/17, 79/29, 193/71, 11251/4139. For example, if p and q are primes and q > 71, and p/q > e, then 193/71 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, A265817, A265818, A265819.

Sequence in context: A165353 A048976 A088546 * A246717 A295706 A265792

Adjacent sequences: A265813 A265814 A265815 * A265817 A265818 A265819

Keyword

nonn,frac

Author

Clark Kimberling, Jan 02 2016

Extensions

More terms from Bert Dobbelaere, Jul 21 2022

Status

approved