login
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.
4
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, 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, 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
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
Christian Elsholtz, Sums Of k Unit Fractions, Trans. Amer. Math. Soc. 353 (2001), 3209-3227.
David Eppstein, Ten Algorithms for Egyptian Fractions, Wolfram Library Archive.
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
Cf. A002966, A073546, A227610 (1/n), A227611 (2/n), A075785 (3/n), A073101 (4/n), A075248 (5/n).
Sequence in context: A137943 A202189 A202183 * A221273 A352607 A202185
KEYWORD
nonn,tabl
AUTHOR
Robert G. Wilson v, Jul 17 2013
STATUS
approved