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A265764
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Denominators of primes-only best approximates (POBAs) to 3; see Comments.
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3
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2, 2, 5, 5, 7, 7, 11, 13, 13, 17, 19, 23, 23, 29, 37, 37, 43, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 103, 103, 113, 127, 127, 137, 139, 149, 163, 163, 167, 167, 173, 181, 191, 193, 197, 199, 211, 227, 233, 239, 251, 257, 257, 263, 269, 271, 277, 293
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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.
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LINKS
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EXAMPLE
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The POBAs for 3 start with 7/2, 5/2, 17/5, 13/5, 23/7, 19/7, 31/11, 41/13, 37/13, 53/17. For example, if p and q are primes and q > 13, then 41/13 is closer to 3 than p/q is.
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MATHEMATICA
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x = 3; 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]] &];
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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