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A265784 Numerators of lower primes-only best approximates (POBAs) to sqrt(5); see Comments. 7
3, 5, 11, 29, 163, 199, 521, 3571, 91283, 150427 (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..10.

EXAMPLE

The lower POBAs to sqrt(5) start with 3/2, 5/3, 11/5, 29/13, 163/73, 199/89, 521/233. For example, if p and q are primes and q > 73, and p/q < sqrt(5), then 163/73 is closer to sqrt(5) than p/q is.

MATHEMATICA

x = Sqrt[5]; 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]   (* A265784 *)

Denominator[tL] (* A265785 *)

Numerator[tU]   (* A265786 *)

Denominator[tU] (* A265787 *)

Numerator[y]    (* A222588 *)

Denominator[y]  (* A265789 *)

CROSSREFS

Cf. A000040, A265759, A265785, A265786, A265787, A265788, A265789.

Sequence in context: A253899 A037152 A084748 * A146243 A262936 A214089

Adjacent sequences:  A265781 A265782 A265783 * A265785 A265786 A265787

KEYWORD

nonn,frac,more

AUTHOR

Clark Kimberling, Dec 23 2015

EXTENSIONS

a(9)-a(10) from Robert Price, Apr 05 2019

STATUS

approved

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Last modified August 9 01:35 EDT 2020. Contains 336310 sequences. (Running on oeis4.)