OFFSET
1,3
COMMENTS
N. J. A. Sloane and R. H. Hardin conjecture a(n) > 0 for n > 1. All of the solutions can be printed by removing the comment symbols from the Mathematica program. For the solution (x,y,z) having the largest z value, see (A075260, A075261, A075262). See A073101 for the 4/n conjecture due to Erdős and Straus.
The conjecture was proved by Thomas Hagedorn and Gary Mulkey. - T. D. Noe, Jan 03 2005
REFERENCES
R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, D11.
LINKS
T. D. Noe, Table of n, a(n) for n=1..499
Thomas R. Hagedorn, A proof of a conjecture on Egyptian fractions, Amer. Math Monthly, 107 (2000), 62-63.
Stan Wagon, Problem of the Week 848: An Odd Egyptian Puzzle
Eric Weisstein's World of Mathematics, Egyptian Fraction
EXAMPLE
a(3)=2 because there are two solutions: 3/7 = 1/3+1/11+1/231 and 3/7 = 1/3+1/15+1/35.
MATHEMATICA
m = 3; For[lst = {}; n = 3, n <= 200, n = n + 2, cnt = 0; xr = n/m; If[IntegerQ[xr], xMin = xr + 1, xMin = Ceiling[xr]]; If[IntegerQ[3xr], xMax = 3xr - 1, xMax = Floor[3xr]]; For[x = xMin, x <= xMax, x++, yr = 1/(m/n - 1/x); If[IntegerQ[yr], yMin = yr + 1, yMin = Ceiling[yr]]; If[IntegerQ[2yr], yMax = 2yr + 1, yMax = Ceiling[2yr]]; For[y = yMin, y <= yMax, y++, zr = 1/(m/n - 1/x - 1/y); If[y > x && zr > y && IntegerQ[zr], z = zr; If[OddQ[x y z], cnt++; (*Print[n, " ", x, " ", y, " ", z]*)]]]]; AppendTo[lst, cnt]]; lst
CROSSREFS
KEYWORD
nice,nonn
AUTHOR
T. D. Noe, Sep 10 2002
EXTENSIONS
More terms from T. D. Noe, Oct 15 2002
STATUS
approved