A075259 Number of solutions (x,y,z) to 3/(2n+1) = 1/x + 1/y + 1/z satisfying 0 < x < y < z and odd x, y, z.

0, 1, 2, 1, 1, 5, 2, 1, 3, 5, 1, 12, 8, 3, 3, 5, 14, 8, 6, 4, 7, 20, 1, 9, 6, 3, 22, 11, 3, 11, 31, 24, 5, 10, 3, 11, 16, 20, 6, 23, 2, 35, 7, 3, 35, 15, 25, 16, 47, 8, 12, 54, 3, 9, 8, 4, 42, 41, 22, 11, 8, 25, 8, 15, 5, 61, 92, 3, 7, 16, 28, 47, 37, 7, 10, 40, 23, 13, 11, 29, 11, 75, 3

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

Cf. A073101, A075260, A075261, A075262.

Sequence in context: A047888 A330964 A128704 * A307877 A259703 A316996

Adjacent sequences: A075256 A075257 A075258 * A075260 A075261 A075262

Keyword

nice,nonn

Author

T. D. Noe, Sep 10 2002

Extensions

More terms from T. D. Noe, Oct 15 2002

Status

approved