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Number of ways to express m/n as Egyptian fractions in just three terms, that is, m/n = 1/x + 1/y + 1/z satisfying 1 <= x <= y <= z and read by antidiagonals.
6

%I #27 Oct 11 2023 15:18:23

%S 3,1,10,1,3,21,0,3,8,28,0,1,3,10,36,0,1,3,6,12,57,0,1,2,3,10,21,42,0,

%T 0,1,4,2,10,17,70,0,0,1,3,3,8,9,28,79,0,0,0,1,3,4,7,20,26,96,0,0,1,1,

%U 2,3,4,10,21,36,62,0,0,0,1,1,7,1,7,6,21,25,160,0,0,0,1,0,3,3,6,12,12,16,57,59

%N Number of ways to express m/n as Egyptian fractions in just three terms, that is, m/n = 1/x + 1/y + 1/z satisfying 1 <= x <= y <= z and read by antidiagonals.

%C See A073101 for the 4/n conjecture due to Erdös and Straus.

%C The first upper diagonal is 10, 8, 6, 2, 4, 1, 2, 1, 2, 0, 3, 0, 0, 1, 0, 0, 1, 0, 1, 0,... .

%C The main diagonal is: 3, 3, 3, 3, 3, 3, ... since 1 = 1/2 + 1/3 + 1/6 = 1/2 + 1/4 + 1/4 = 1/3 + 1/3 + 1/3. See A002966(3).

%C The first lower diagonal is 1, 3, 3, 4, 3, 7, 3, 5, 4, 6, 3, 10, 3, 6, 6, 6, 3, 9, 3, 9, ... .

%C The antidiagonal sum is 3, 11, 25, 39, 50, 79, 79, 104, 131, 157, 140, 229, 169, 220, 295, 282, ... .

%H Christian Elsholtz, <a href="http://www.ams.org/tran/2001-353-08/S0002-9947-01-02782-9/home.html">Sums Of k Unit Fractions</a>

%H David Eppstein, <a href="http://www.ics.uci.edu/~eppstein/numth/egypt/intro.html">Algorithms for Egyptian Fractions</a>

%H David Eppstein, <a href="http://library.wolfram.com/infocenter/Articles/2926/">Ten Algorithms for Egyptian Fractions</a>

%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html">Egyptian Fractions</a>

%H Oakland University, <a href="http://www.oakland.edu/enp/">The Erdős Number Project</a>

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

%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>

%e ../n

%e m/ 1...2...3...4...5...6...7...8...9..10..11...12..13...14...15 =Allocation nbr.

%e .1 3..10..21..28..36..57..42..70..79..96..62..160..59..136..196 A004194

%e .2 1...3...8..10..12..21..17..28..26..36..25...57..20...42...81 A226641

%e .3 1...3...3...6..10..10...9..20..21..21..16...28..11...33...36 A226642

%e .4 0...1...3...3...2...8...7..10...6..12...9...21...4...17...39 A192787

%e .5 0...1...2...4...3...4...4...7..12..10...3...17...6...21...21 A226644

%e .6 0...1...1...3...3...3...1...6...8..10...7...10...1....9...12 A226645

%e .7 0...0...1...1...2...7...3...2...3...5...2...13...8...10....9 n/a

%e .8 0...0...0...1...1...3...3...3...1...2...0....8...3....7...19 n/a

%e .9 0...0...1...1...0...3...2...5...3...2...0....6...2....4...10 n/a

%e 10 0...0...0...1...1...2...0...4...4...3...0....4...1....4....8 n/a

%e Triangle (by antidiagonals):

%e {3},

%e {1, 10},

%e {1, 3, 21},

%e {0, 3, 8, 28},

%e {0, 1, 3, 10, 36},

%e {0, 1, 3, 6, 12, 57},

%e ...

%t f[m_, n_] := Length@ Solve[m/n == 1/x + 1/y + 1/z && 1 <= x <= y <= z, {x, y, z}, Integers]; Table[f[n, m - n + 1], {m, 12}, {n, m, 1, -1}] // Flatten

%Y Cf. A227612, A226640, A226641, A226642, A192787, A226644, A226645.

%K nonn,tabl

%O 1,1

%A _Allan C. Wechsler_ and _Robert G. Wilson v_, Aug 17 2013