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A265775 Denominators of upper primes-only best approximates (POBAs) to sqrt(2); see Comments. 7
2, 13, 37, 43, 139, 149, 313, 347, 593, 743, 883, 1009, 2617, 12269, 15731, 37879, 43789, 90533 (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..18.

EXAMPLE

The upper POBAs to sqrt(2) start with 3/2, 19/13, 53/37, 61/43, 197/139, 211/149. For example, if p and q are primes and q > 139, and p/q > sqrt(2), then 197/139 is closer to sqrt(2) than p/q is.

MATHEMATICA

x = Sqrt[2]; z = 200; 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, A265776/A265777 *)

Numerator[tL]   (* A265772 *)

Denominator[tL] (* A265773 *)

Numerator[tU]   (* A265774 *)

Denominator[tU] (* A265775 *)

Numerator[y]    (* A265776 *)

Denominator[y]  (* A265777 *)

CROSSREFS

Cf. A000040, A265759, A265772, A265773, A265774, A265776, A265777.

Sequence in context: A034011 A085497 A320515 * A291205 A005113 A239196

Adjacent sequences:  A265772 A265773 A265774 * A265776 A265777 A265778

KEYWORD

nonn,frac,more

AUTHOR

Clark Kimberling, Dec 20 2015

EXTENSIONS

a(14)-a(18) from Robert Price, Apr 05 2019

STATUS

approved

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Last modified February 24 03:22 EST 2020. Contains 332195 sequences. (Running on oeis4.)