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A265778 Numerators of lower primes-only best approximates (POBAs) to sqrt(3); see Comments. 7
3, 5, 19, 71, 601, 1997, 2579, 3691, 75533, 167543, 175649 (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..11.

EXAMPLE

The lower POBAs to sqrt(3) start with 3/2, 5/3, 19/11, 71/41, 601/347. For example, if p and q are primes and q > 347, and p/q < sqrt(3), then 601/347 is closer to sqrt(3) than p/q is.

MATHEMATICA

x = Sqrt[3]; 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, A265782/A265783 *)

Numerator[tL]   (* A265778 *)

Denominator[tL] (* A265779 *)

Numerator[tU]   (* A265780 *)

Denominator[tU] (* A265781 *)

Numerator[y]    (* A262582 *)

Denominator[y]  (* A265783 *)

CROSSREFS

Cf. A000040, A265759, A265779, A265780, A265781, A265782, A265783.

Sequence in context: A106918 A182357 A181519 * A257866 A251617 A055452

Adjacent sequences:  A265775 A265776 A265777 * A265779 A265780 A265781

KEYWORD

nonn,frac,more

AUTHOR

Clark Kimberling, Dec 20 2015

EXTENSIONS

a(9)-a(11) from Robert Price, Apr 05 2019

STATUS

approved

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Last modified November 24 18:11 EST 2020. Contains 338616 sequences. (Running on oeis4.)