OFFSET
1,3
COMMENTS
The main diagonal is 1, 1, 1, 1, 1, 1, 1, ..., ; i.e., 1 = 1/2 + 1/3 + 1/6.
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Christian Elsholtz, Sums Of k Unit Fractions, Trans. Amer. Math. Soc. 353 (2001), 3209-3227.
David Eppstein, Algorithms for Egyptian Fractions
David Eppstein, Ten Algorithms for Egyptian Fractions, Wolfram Library Archive.
Ron Knott Egyptian Fractions
Oakland University, The Erdős Number Project
Eric Weisstein's World of Mathematics, Egyptian Fraction
EXAMPLE
m\n| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---+------------------------------------------------------------
1 | 1 6 15 22 30 45 36 62 69 84 56 142 53 124 178 A227610
2 | 0 1 5 6 9 15 14 22 21 30 22 45 17 36 72 A227611
3 | 0 1 1 3 7 6 6 16 15 15 13 22 8 27 30 A075785
4 | 0 0 1 1 2 5 5 6 4 9 7 15 4 14 33 A073101
5 | 0 0 1 2 1 1 3 5 9 6 3 12 5 18 15 A075248
6 | 0 0 0 1 1 1 1 3 5 7 5 6 1 6 9 n/a
7 | 0 0 0 1 1 4 1 2 2 2 2 9 6 6 7 n/a
8 | 0 0 0 0 1 1 1 1 1 2 0 5 3 5 15 n/a
9 | 0 0 0 0 0 1 1 3 1 1 0 3 1 2 7 n/a
10 | 0 0 0 0 0 1 0 2 2 1 0 1 1 3 5 n/a
.
Antidiagonals are {1}, {0, 6}, {0, 1, 15}, {0, 1, 5, 22}, {0, 0, 1, 6, 30}, {0, 0, 1, 3, 9, 45}, ...
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Robert G. Wilson v, Jul 17 2013
STATUS
approved