
COMMENTS

All denominators in the expansion 1 = 1/x_1 + ... + 1/x_n are bounded by A000058(n1), i.e., 0 < x_1 <= ... <= x_n < A000058(n1). Furthermore, for a fixed n, x_i <= (n+1i)*(A000058(i1)1). [Max Alekseyev, Oct 11 2012]
From R. J. Mathar, May 06 2010: (Start)
This is the leading edge of the triangle A156869. This is also the row n=1 of an array T(n,m) which counts the number of ways to write 1/n as a sum over m (not necessarily distinct) unit fractions:
1.1...3...14....147....3462..294314
1.2..10..108...2892..270332........
1.2..21..339..17253................
1.3..28..694..51323................
T(.,2) = A018892. T(.,3) = A004194. T(.,4) = A020327, T(.,5) = A020328. T(2,6) is computed by D. S. McNeil, who conjectures that the 2nd row is A003167. (End)
If on the other hand, all x_k must be unique, see A006585.  Robert G. Wilson v, Jul 17 2013


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, D11.
D. Singmaster, ``The number of representations of one as a sum of unit fractions,'' unpublished manuscript, 1972.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


EXAMPLE

For n=3 the 3 solutions are {2,3,6}, {2,4,4}, {3,3,3}.
For n=4 the solutions are: {2,3,7,42}, {2,3,8,24}, {2,3,9,18}, {2,3,10,15}, {2,3,12,12}, {2,4,5,20}, {2,4,6,12}, {2,4,8,8}, {2,5,5,10}, {2,6,6,6}, {3,3,4,12}, {3,3,6,6}, {3,4,4,6}, {4,4,4,4} [Neven Juric, May 14 2008]
