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A265800
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Numerators of primes-only best approximates (POBAs) to the golden ratio, tau; see Comments.
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8
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5, 3, 5, 11, 31, 37, 47, 157, 571, 911, 1021, 1487, 2351, 3571, 24709, 25463, 69247, 80803
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OFFSET
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1,1
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COMMENTS
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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.
How is this related to A165572? - R. J. Mathar, Jan 10 2016
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LINKS
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Table of n, a(n) for n=1..18.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A000040, A265759, A265796, A265797, A265798, A265799, A265801.
Sequence in context: A185046 A114740 A128008 * A144386 A256787 A265782
Adjacent sequences: A265797 A265798 A265799 * A265801 A265802 A265803
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KEYWORD
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nonn,frac,more
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AUTHOR
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Clark Kimberling, Dec 29 2015
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EXTENSIONS
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a(15)-a(18) from Robert Price, Apr 06 2019
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STATUS
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approved
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