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 A265794 Numerators of primes-only best approximates (POBAs) to sqrt(8); see Comments. 7
 7, 5, 13, 19, 31, 167, 359, 461, 659, 1847, 2803, 4517, 32377, 35839, 199373 (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(8) start with 7/2, 5/2, 13/5, 19/7, 31/11, 167/59, 359/127, 461/163, 659/233. For example, if p and q are primes and q > 59, then 167/59 is closer to sqrt(8) than p/q is. MATHEMATICA x = Sqrt[8]; 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, A265794/A265795 *) Numerator[tL] (* A265790 *) Denominator[tL] (* A265791 *) Numerator[tU] (* A265792 *) Denominator[tU] (* A265793 *) Numerator[y] (* A265794 *) Denominator[y] (* A265795 *) CROSSREFS Cf. A000040, A265759, A265790, A265791, A265792, A265793, A265795. Sequence in context: A107471 A332768 A107323 * A265818 A046557 A298722 Adjacent sequences: A265791 A265792 A265793 * A265795 A265796 A265797 KEYWORD nonn,frac,more AUTHOR Clark Kimberling, Dec 29 2015 EXTENSIONS a(13)-a(15) from Robert Price, Apr 06 2019 STATUS approved

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Last modified September 15 13:51 EDT 2024. Contains 375938 sequences. (Running on oeis4.)