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 A265765 Numerators of primes-only best approximates (POBAs) to 4; see Comments. 2
 11, 7, 13, 11, 19, 29, 43, 53, 67, 149, 163, 173, 211, 269, 283, 293, 317, 331, 389, 509, 523, 547, 557, 653, 691, 773, 787, 797, 907, 1051, 1109, 1123, 1171, 1229, 1493, 1531, 1637, 1723, 1733, 1867, 1949, 1997, 2011, 2083, 2251, 2309, 2347, 2371, 2467 (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..49. EXAMPLE The POBAs for 4 start with 11/2, 7/2, 13/3, 11/3, 19/5, 29/7, 43/11, 53/13, 67/17. For example, if p and q are primes and q > 13, then 53/13 is closer to 3 than p/q is. MATHEMATICA x = 4; 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, A265765/A120639 *) Numerator[tL] (* A162857 *) Denominator[tL] (* A062737 *) Numerator[tU] (* A090866 *) Denominator[tU] (* A023212 *) Numerator[y] (* A265765 *) Denominator[y] (* A120639 *) CROSSREFS Cf. A000040, A023212, A062737, A090866, A120639, A162857. Sequence in context: A144262 A110093 A282345 * A187563 A089487 A166521 Adjacent sequences: A265762 A265763 A265764 * A265766 A265767 A265768 KEYWORD nonn,frac AUTHOR Clark Kimberling, Dec 18 2015 STATUS approved

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Last modified August 11 19:17 EDT 2024. Contains 375073 sequences. (Running on oeis4.)