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A265814 Numerators of lower primes-only best approximates (POBAs) to e; see Comments. 7
5, 13, 19, 307, 443, 617, 2237, 2411, 2971, 5923, 7043, 7603, 11887, 12659, 15361, 24103, 75223, 89021, 128273 (list; graph; refs; listen; history; text; internal format)
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..19.

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,more

AUTHOR

Clark Kimberling, Jan 02 2016

STATUS

approved

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Last modified October 18 14:52 EDT 2019. Contains 328161 sequences. (Running on oeis4.)