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A265780
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Numerators of upper primes-only best approximates (POBAs) to sqrt(3); see Comments.
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7
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5, 7, 11, 13, 23, 83, 103, 127, 137, 227, 809, 1093, 1571, 4273, 5333, 16141, 20627, 41519, 56813, 111913
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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 an upper primes-only best approximate, and we write "p/q is in U(x)", if p'/q < x < p/q < u/v for all primes u and v such that v < q, where p' is greatest prime < p in case p >= 3.
Let q(1) = 2 and let p(1) be the least prime >= x. The sequence U(x) follows inductively: for n >= 1, let q(n) is the least prime q such that x < p/q < p(n)/q(n) for some prime p. Let q(n+1) = q and let p(n+1) be the least prime p such that x < p/q < p(n)/q(n).
For a guide to POBAs, lower POBAs, and upper POBAs, see A265759.
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LINKS
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FORMULA
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The upper POBAs to sqrt(3) start with 5/2, 7/3, 11/5, 13/7, 23/13, 83/47, 103/59. For example, if p and q are primes and q > 47, and p/q > sqrt(3), then 83/47 is closer to sqrt(3) than p/q is.
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MATHEMATICA
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x = Sqrt[3]; 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]] &];
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CROSSREFS
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KEYWORD
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nonn,frac,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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