This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(16) = 32640 because 16/17 = 1/2 + 1/3 + 1/10 + 1/128 + 1/32640.
a100695 = last . a247765_row -- Reinhard Zumkeller, Sep 25 2014
a[n_] := Module[{m = n/(n+1)}, While[Numerator[m]>1, m = m-1/Ceiling[1/m]]; 1/m]; Array[a, 100] (* Jean-François Alcover, Mar 12 2019, after M. F. Hasler *)
a(n)={n/=(n+1);while(numerator(n)>1,n-=1/ceil(1/n));1/n} \\ M. F. Hasler, Sep 24 2014
. 1: 2 . 2: 2, 6 . 3: 2, 4 . 4: 2, 4, 20 . 5: 2, 3 . 6: 2, 3, 42 . 7: 2, 3, 24 . 8: 2, 3, 18 . 9: 2, 3, 15 . 10: 2, 3, 14, 231 . 11: 2, 3, 12 . 12: 2, 3, 12, 156 . 13: 2, 3, 11, 231 . 14: 2, 3, 10 . 15: 2, 3, 10, 240 . 16: 2, 3, 10, 128, 32640 . 17: 2, 3, 9 . 18: 2, 3, 9, 342 . 19: 2, 3, 9, 180 . 20: 2, 3, 9, 126
import Data.Ratio ((%), numerator, denominator)
a247765 n k = a247765_tabf !! (n-1) !! (k-1)
a247765_tabf = map a247765_row [1..]
a247765_row n = f (map recip [2..]) (n % (n + 1)) where
f es x | numerator x == 1 = [denominator x]
| otherwise = g es
where g (u:us) | u <= x = (denominator u) : f us (x - u)
| otherwise = g us
For n = 1, 1-1/(2n+1) = 2/3 = 1/3 + 1/5 + 1/9 + 1/45 has 4 fractions in the representation, thus a(1) = 4.
odd[n_]:=If[OddQ[n],n,n+1];a={};For[n=0,n<100,n++;lst={};k=2n/(2n+1);s1=0; While[k>0,s2=odd[Ceiling[1/k]]; If[s2==s1,s2+=2]; AppendTo[lst, s2]; k=k-1/s2; s1=s2];a=AppendTo[a,Length[lst]]];a
a={};dmax=0;n=1;While[Length[a]<40,denom=0; k=n/(n+1); While[ k>0,denom=Ceiling[1/k]; k=k-1/denom]; If[denom>dmax,dmax=denom;a=AppendTo[a,n]];n++];a
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