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A265816 Numerators of upper primes-only best approximates (POBAs) to e; see Comments. 7
7, 17, 23, 31, 47, 79, 193, 11251, 15149, 17291, 25261, 46643, 49171 (list; graph; refs; listen; history; text; internal format)
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..13.

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,more

AUTHOR

Clark Kimberling, Jan 02 2016

STATUS

approved

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Last modified July 20 01:17 EDT 2019. Contains 325168 sequences. (Running on oeis4.)