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.
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
Examples
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
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Antidiagonals are {1}, {0, 6}, {0, 1, 15}, {0, 1, 5, 22}, {0, 0, 1, 6, 30}, {0, 0, 1, 3, 9, 45}, ...
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
- Index entries for sequences related to Egyptian fractions
Crossrefs
Programs
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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}]
Comments