A227612 - OEIS

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 #30 Feb 16 2025 08:33:20

%S 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 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 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 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 The main diagonal is 1, 1, 1, 1, 1, 1, 1, ..., ; i.e., 1 = 1/2 + 1/3 + 1/6.

%H G. C. Greubel, <a href="/A227612/b227612.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H 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 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>, Wolfram Library Archive.

%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="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>

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

%e m\n| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

%e ---+------------------------------------------------------------

%e 1 | 1 6 15 22 30 45 36 62 69 84 56 142 53 124 178 A227610

%e 2 | 0 1 5 6 9 15 14 22 21 30 22 45 17 36 72 A227611

%e 3 | 0 1 1 3 7 6 6 16 15 15 13 22 8 27 30 A075785

%e 4 | 0 0 1 1 2 5 5 6 4 9 7 15 4 14 33 A073101

%e 5 | 0 0 1 2 1 1 3 5 9 6 3 12 5 18 15 A075248

%e 6 | 0 0 0 1 1 1 1 3 5 7 5 6 1 6 9 n/a

%e 7 | 0 0 0 1 1 4 1 2 2 2 2 9 6 6 7 n/a

%e 8 | 0 0 0 0 1 1 1 1 1 2 0 5 3 5 15 n/a

%e 9 | 0 0 0 0 0 1 1 3 1 1 0 3 1 2 7 n/a

%e 10 | 0 0 0 0 0 1 0 2 2 1 0 1 1 3 5 n/a

%e .

%e Antidiagonals are {1}, {0, 6}, {0, 1, 15}, {0, 1, 5, 22}, {0, 0, 1, 6, 30}, {0, 0, 1, 3, 9, 45}, ...

%t 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 Cf. A002966, A073546, A227610 (1/n), A227611 (2/n), A075785 (3/n), A073101 (4/n), A075248 (5/n).

%K nonn,tabl,changed

%O 1,3

%A _Robert G. Wilson v_, Jul 17 2013