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A079940
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Greedy fractional multiples of 1/e: a(1)=1, sum_{n>0} fractional_part(a(n)/e) = 1.
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5
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1, 3, 4, 11, 87, 193, 386, 579, 1457, 23225, 49171, 98342, 147513, 196684, 566827, 13580623, 28245729, 56491458, 84737187, 112982916
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OFFSET
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1,2
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COMMENTS
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The n-th greedy fractional multiple of x is the smallest integer m that does not cause sum(k=1..n,frac(m*x)) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of 1/e.
After a(20), there is only 109305220 - 297122396/e = ~1.06317354345346734...*10^-8 to work with.
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LINKS
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EXAMPLE
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a(4) = 11 since frac(1x) + frac(3x) + frac(4x) + frac(11x) < 1, while frac(1x) + frac(3x) + frac(4x) + frac(k*x) > 1 for all k>4 and k<11.
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MAPLE
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Digits := 100: a := []: s := 0: x := 1.0/exp(1.0): for n from 1 to 1000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a), n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;
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MATHEMATICA
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a[1] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 1, ps = Plus @@ Table[ FractionalPart[ a[i]*E^-1], {i, n - 1}]}, While[ ps + FractionalPart[k*E^-1] > 1, k++ ]; a[n] = k]; Do[ Print[ a[n]], {n, 20}] (* Robert G. Wilson v, Nov 03 2004 *)
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CROSSREFS
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KEYWORD
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more,nonn
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AUTHOR
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EXTENSIONS
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More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 30 2003
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STATUS
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approved
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