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

1,3

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

Index entries for sequences related to Egyptian fractions

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

Adjacent sequences: A227609 A227610 A227611 * A227613 A227614 A227615

Keyword

nonn,tabl

Author

Robert G. Wilson v, Jul 17 2013

Status

approved