A265814 Numerators of lower primes-only best approximates (POBAs) to e; see Comments.

5, 13, 19, 307, 443, 617, 2237, 2411, 2971, 5923, 7043, 7603, 11887, 12659, 15361, 24103, 75223, 89021, 128273, 283949, 423299, 1169027, 1587077, 1830211, 3062207, 5080939, 8481901, 9366979, 22675801, 67090433, 71625049, 191016521, 211670869, 221578729, 244402043, 428023867, 1451377009

Offset

1,1

Comments

Suppose that x > 0. A fraction p/q of primes is a lower primes-only best approximate, and we write "p/q is in L(x)", if u/v < p/q < x < p'/q for all primes u and v such that v < q, where p' is least prime > p.

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

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

Links

Table of n, a(n) for n=1..37.

Example

The lower POBAs to e; start with 5/2, 13/5, 19/7, 307/113, 443/163, 617/227, 2237/823. For example, if p and q are primes and q > 823, and p/q < e, then 2237/823 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, A265815, A265816, A265817, A265818, A265819.

Sequence in context: A232655 A175866 A227500 * A087714 A055045 A213741

Adjacent sequences: A265811 A265812 A265813 * A265815 A265816 A265817

Keyword

nonn,frac

Author

Clark Kimberling, Jan 02 2016

Extensions

More terms from Bert Dobbelaere, Jul 21 2022

Status

approved