A075259 - OEIS

%I #18 Nov 16 2014 03:10:03

%S 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,

%T 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,

%U 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

%N 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.

%C _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.

%C The conjecture was proved by Thomas Hagedorn and Gary Mulkey. - _T. D. Noe_, Jan 03 2005

%D R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, D11.

%H T. D. Noe, <a href="/A075259/b075259.txt">Table of n, a(n) for n=1..499</a>

%H Thomas R. Hagedorn, <a href="http://www.jstor.org/stable/2589381">A proof of a conjecture on Egyptian fractions</a>, Amer. Math Monthly, 107 (2000), 62-63.

%H Stan Wagon, <a href="http://compgeom.cs.uiuc.edu/~jeffe/wagon/s848.html">Problem of the Week 848: An Odd Egyptian Puzzle</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>

%e 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.

%t 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

%Y Cf. A073101, A075260, A075261, A075262.

%K nice,nonn

%O 1,3

%A _T. D. Noe_, Sep 10 2002

%E More terms from _T. D. Noe_, Oct 15 2002