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A227612 Table read by antidiagonals: Number of ways m/n can be expressed as the sum of three distinct unit fractions, i.e., m/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and read by antidiagonals.

%I A227612 #30 Feb 16 2025 08:33:20
%S A227612 1,0,6,0,1,15,0,1,5,22,0,0,1,6,30,0,0,1,3,9,45,0,0,1,1,7,15,36,0,0,0,
%T A227612 2,2,6,14,62,0,0,0,1,1,5,6,22,69,0,0,0,1,1,1,5,16,21,84,0,0,0,0,1,1,3,
%U A227612 6,15,30,56,0,0,0,0,1,4,1,5,4,15,22,142,0,0,0,0,0,1,1,3,9,9,13,45,53
%N A227612 Table read by antidiagonals: Number of ways m/n can be expressed as the sum of three distinct unit fractions, i.e., m/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and read by antidiagonals.
%C A227612 The main diagonal is 1, 1, 1, 1, 1, 1, 1, ..., ; i.e., 1 = 1/2 + 1/3 + 1/6.
%H A227612 G. C. Greubel, <a href="/A227612/b227612.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A227612 Christian Elsholtz, <a href="https://doi.org/10.1090/S0002-9947-01-02782-9">Sums Of k Unit Fractions</a>, Trans. Amer. Math. Soc. 353 (2001), 3209-3227.
%H A227612 David Eppstein, <a href="http://www.ics.uci.edu/~eppstein/numth/egypt/intro.html">Algorithms for Egyptian Fractions</a>
%H A227612 David Eppstein, <a href="http://library.wolfram.com/infocenter/Articles/2926/">Ten Algorithms for Egyptian Fractions</a>, Wolfram Library Archive.
%H A227612 Ron Knott <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html">Egyptian Fractions</a>
%H A227612 Oakland University, <a href="http://www.oakland.edu/enp/">The Erdős Number Project</a>
%H A227612 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A227612 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A227612   m\n| 1  2   3   4   5   6   7   8   9  10  11   12  13   14   15
%e A227612   ---+------------------------------------------------------------
%e A227612    1 | 1  6  15  22  30  45  36  62  69  84  56  142  53  124  178  A227610
%e A227612    2 | 0  1   5   6   9  15  14  22  21  30  22   45  17   36   72  A227611
%e A227612    3 | 0  1   1   3   7   6   6  16  15  15  13   22   8   27   30  A075785
%e A227612    4 | 0  0   1   1   2   5   5   6   4   9   7   15   4   14   33  A073101
%e A227612    5 | 0  0   1   2   1   1   3   5   9   6   3   12   5   18   15  A075248
%e A227612    6 | 0  0   0   1   1   1   1   3   5   7   5    6   1    6    9  n/a
%e A227612    7 | 0  0   0   1   1   4   1   2   2   2   2    9   6    6    7  n/a
%e A227612    8 | 0  0   0   0   1   1   1   1   1   2   0    5   3    5   15  n/a
%e A227612    9 | 0  0   0   0   0   1   1   3   1   1   0    3   1    2    7  n/a
%e A227612   10 | 0  0   0   0   0   1   0   2   2   1   0    1   1    3    5  n/a
%e A227612 .
%e A227612 Antidiagonals are {1}, {0, 6}, {0, 1, 15}, {0, 1, 5, 22}, {0, 0, 1, 6, 30}, {0, 0, 1, 3, 9, 45}, ...
%t A227612 f[m_, n_] := Length@ Solve[m/n == 1/x + 1/y + 1/z && 0 < x < y < z, {x, y, z}, Integers]; Table[ f[n, m - n + 1], {m, 12}, {n, m, 1, -1}]
%Y A227612 Cf. A002966, A073546, A227610 (1/n), A227611 (2/n), A075785 (3/n), A073101 (4/n), A075248 (5/n).
%K A227612 nonn,tabl
%O A227612 1,3
%A A227612 _Robert G. Wilson v_, Jul 17 2013