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A265777 Denominators of primes-only best approximates (POBAs) to sqrt(2); see Comments. 7
2, 2, 5, 29, 691, 773, 971, 1009, 2617, 6079, 15731, 27073, 37879, 43789, 55609 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..15.

EXAMPLE

The POBAs to sqrt(2) start with 2/2, 3/2, 7/5, 41/29, 977/691, 1093/773, 1373/971, 1427/1009. For example, if p and q are primes and q > 29, then 41/29 is closer to sqrt(2) than p/q is.

MATHEMATICA

x = Sqrt[2]; z = 800; 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, A265775, A265776.

Sequence in context: A014566 A259861 A293264 * A076658 A218270 A218122

Adjacent sequences:  A265774 A265775 A265776 * A265778 A265779 A265780

KEYWORD

nonn,frac,more

AUTHOR

Clark Kimberling, Dec 20 2015

EXTENSIONS

a(11)-a(15) from Robert Price, Apr 05 2019

STATUS

approved

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Last modified February 23 00:28 EST 2020. Contains 332157 sequences. (Running on oeis4.)