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 A265801 Denominators of primes-only best approximates (POBAs) to the golden ratio, tau; see Comments. 12
 2, 2, 3, 7, 19, 23, 29, 97, 353, 563, 631, 919, 1453, 2207, 15271, 15737, 42797, 49939 (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. Is this related to A165571? - R. J. Mathar, Jan 10 2016 LINKS EXAMPLE The POBAs to tau start with 5/2, 3/2, 5/3, 11/7, 31/19, 37/23, 47/29, 157/97, 571/353, 911/563. For example, if p and q are primes and q > 29, then 47/29 is closer to tau than p/q is. MATHEMATICA x = GoldenRatio; 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, A265800/A265801 *) Numerator[tL]   (* A265796 *) Denominator[tL] (* A265797 *) Numerator[tU]   (* A265798 *) Denominator[tU] (* A265799 *) Numerator[y]    (* A265800 *) Denominator[y]  (* A265801 *) CROSSREFS Cf. A000040, A265759, A265796, A265797, A265798, A265799, A265800. Sequence in context: A117387 A113842 A032161 * A098738 A291742 A083701 Adjacent sequences:  A265798 A265799 A265800 * A265802 A265803 A265804 KEYWORD nonn,frac,more AUTHOR Clark Kimberling, Jan 02 2016 EXTENSIONS a(15)-a(18) from Robert Price, Apr 06 2019 STATUS approved

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Last modified July 22 10:41 EDT 2019. Contains 325219 sequences. (Running on oeis4.)