Search: id:A075259
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%I A075259
%S A075259 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 A075259 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,
%U A075259 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 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.
%C A075259 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 Erdos and Straus.
%C A075259 The conjecture was proved by Thomas Hagedorn and Gary Mulkey. - T. D. 
               Noe (noe(AT)sspectra.com), Jan 03 2005
%D A075259 R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third 
               Edition, 2004, D11.
%D A075259 Thomas R. Hagedorn, A proof of a conjecture on Egyptian fractions, Amer. 
               Math Monthly, 107 (2000), 62-63.
%H A075259 T. D. Noe, <a href="b075259.txt">Table of n, a(n) for n=1..499</a>
%H A075259 Stan Wagon, <a href="http://compgeom.cs.uiuc.edu/~jeffe/wagon/s848.html">
               Problem of the Week 848: An Odd Egyptian Puzzle</a>
%H A075259 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EgyptianFraction.html">Egyptian Fraction</a>
%e A075259 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 A075259 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 A075259 Cf. A073101, A075260, A075261, A075262.
%Y A075259 Sequence in context: A106270 A047888 A128704 this_sequence A003570 A011281 
               A100398
%Y A075259 Adjacent sequences: A075256 A075257 A075258 this_sequence A075260 A075261 
               A075262
%K A075259 nice,nonn
%O A075259 1,3
%A A075259 T. D. Noe (noe(AT)sspectra.com), Sep 10 2002
%E A075259 More terms from T. D. Noe (noe(AT)sspectra.com), Oct 15 2002

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