```nvec ← P ##.efract Q                        ⍝ Egyptian fractions.

Any  rational number in the range 0<⍵≤1 may be expressed as the sum of the reci-
procal of a finite vector of distinct whole numbers. Such sequences of reciproc-
als are called "Egyptian fractions".

For example:

2       1       1
---  =  ---  +  ---    =    +/ ÷2 6
3       2       6

9       1       1       1
---  =  ---  +  ---  +  ---    =    +/ ÷2 3 15
10       2       3      15

355      1       1        1         1              1
---  =  ---  +  ---  +  ----  +  -------  +  -------------   =   +/ ÷2 29 ..
452      2      29      1093     1790881     6414507721441

NB: a given rational number has any number of Egyptian fraction expansions.

Ref: http://en.wikipedia.org/wiki/Egyptian_fraction

Veli-Matti Jantunen's function [efract] returns an integer vector such that:

(⍺÷⍵) = +/÷ ⍺ efract ⍵          ⍝ quotient = sum of reciprocals.
{⍵≡∪⍵}  ⍺ efract ⍵              ⍝ reciprocals are unique.

For example:

9 efract 10                 ⍝ Egyptian fractions for 9÷10
2 3 15

rational +/÷ 2 3 15         ⍝ check result.
9 10

Technical notes:

There are many ways to find Egyptian fractions; Veli-Matti suggests:

efract←{⎕ML←3
(,⍵){
0∊⍴⍵:⍺[⍋⍺]
x←{⍵ 1×⍵+1}↑⍵
n←2×¯1++/⍵=↑⍵
(⍺,x)∇(n⍴x),⍵~↑⍵
}(⍺-1)/⍵
}

efract←{⎕ML←3           ⍝ Egyptian fractions: Splitting method

(,⍵){
0∊⍴⍵:⍺[⍋⍺]
q←↑⍵ ⋄ x←q 1×q+1 ⋄ b←(x∊⍺)<≠\x
(⍺,b/x)∇((~b)/x),((2×¯1++/⍵=q)⍴x),⍵~q
}(⍺-1)/⍵
}

efract←{                ⍝ Egyptian fractions: Golomb algorithm
(⎕ML ⎕IO)←3 1

⍬{
(p q)←⍵÷(↑⍵){⍵=0:|⍺ ⋄ ⍵ ∇ ⍵|⍺}↑⌽⍵
p=1:{⍵[⍋⍵]}⍺,q

r←(0=p|1+q×⍳q-1)⍳1
s←(1+q×r)÷p
(⍺,s×q)∇ r s
}⍺ ⍵
}

efract←{                ⍝ Egyptian fractions: Binary method
(⎕ML ⎕IO)←3 1

∆G←{⍵=0:|⍺ ⋄ ⍵ ∇ ⍵|⍺}
(p q)←⍺ ⍵÷⍺ ∆G ⍵ ⋄ p=1:q

n←⌈2⍟q ⋄ b←2*n
r←⌊(p×b)÷q ⋄ s←(p×b)-r×q

(a c)←{2*((n⍴2)⊤⍵)/⌽¯1+⍳n}¨r s
(a,c){⍵÷⍺ ∆G¨⍵}((⍴a),⍴c)/b×1 q
}

Examples:

3 efract 7                          ⍝ Egyptian fraction rep of 3÷7
3 11 231

rational +/÷ 3 efract 7             ⍝ round-trip to check.
3 7

rational +/÷ ⎕← 355 efract 452      ⍝ (coarse approximation to ○÷4).
2 4 29 1093 1790881 6.414507721E12
355 452

2 3 5 7 ∘.efract 11 13 17 19        ⍝ ... some more.
┌──────┬─────────┬─────────┬─────────┐
│6 66  │7 91     │9 153    │10 190   │
├──────┼─────────┼─────────┼─────────┤
│4 44  │5 33 2145│6 102    │7 67 8911│
├──────┼─────────┼─────────┼─────────┤
│3 9 99│3 20 780 │4 23 1564│4 76     │
├──────┼─────────┼─────────┼─────────┤
│2 8 88│2 26     │3 13 663 │3 29 1653│
└──────┴─────────┴─────────┴─────────┘