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

Author

Clark Kimberling, Jan 02 2016

Keywords

• nonn
• frac

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.

Examples

			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.
		

Crossrefs

Cf. A000040, A265759, A265814, A265815, 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