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 A265818 Numerators of primes-only best approximates (POBAs) to e; see Comments. 7
 7, 5, 13, 19, 193, 7043, 7603, 11251, 15149, 15361, 17291, 24103, 46643, 49171, 3062207, 5080939, 8481901, 8823377, 22675801, 63342553, 67090433, 71625049, 142362299, 221578729, 244402043, 428023867, 1293881119, 1587183239, 2095606361, 3221097589, 3905501983, 4072807391, 14649723833 (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 POBA's. 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. Many terms of A265806 are also terms of A265801 (denominators of POBAs to tau). LINKS Table of n, a(n) for n=1..33. EXAMPLE The POBAs to Pi start with 7/2, 5/2, 13/5, 19/7, 193/71, 7043/2591, 7603/2797. For example, if p and q are primes and q > 2591, then 7043/2591 is closer to e than p/q is. MATHEMATICA x = E; 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, A265818/A265819 *) Numerator[tL] (* A265814 *) Denominator[tL] (* A265815 *) Numerator[tU] (* A265816 *) Denominator[tU] (* A265817 *) Numerator[y] (* A265818 *) Denominator[y] (* A265819 *) CROSSREFS Cf. A000040, A265759, A265814, A265815, A265816, A265817, A265819. Sequence in context: A332768 A107323 A265794 * A046557 A298722 A299556 Adjacent sequences: A265815 A265816 A265817 * A265819 A265820 A265821 KEYWORD nonn,frac AUTHOR Clark Kimberling, Jan 06 2016 EXTENSIONS More terms from Bert Dobbelaere, Jul 21 2022 STATUS approved

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Last modified August 4 22:06 EDT 2024. Contains 374934 sequences. (Running on oeis4.)