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

Author

Clark Kimberling, Jan 02 2016

Keywords

• nonn
• frac

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.

Examples

			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.
		

Crossrefs

Cf. A000040, A265759, A265815, A265816, A265817, A265818, A265819.

Programs

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 *)

Extensions

More terms from Bert Dobbelaere, Jul 21 2022