|
|
COMMENTS
| All denominators in the expansion 1 = 1/x_1 + ... 1/x_n are bounded by the n-th term of Sylvester's sequence: 0<x_1<=...<=x_n < A000058(n) - Max Alekseyev (maxale(AT)gmail.com), Dec 30 2003
Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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)
|
|
|
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D. Singmaster, ``The number of representations of one as a sum of unit fractions,'' unpublished manuscript, 1972.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} (from Neven Juric, May 14 2008)
|
