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Special Natural Numbers (Posted on 2007-10-14) Difficulty: 3 of 5
Find all natural numbers which can be written in the form (a+b+c)2/(abc) where a,b,c are also natural numbers.

See The Solution Submitted by Praneeth    
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re(2): Some were discovered.... | Comment 4 of 8 |
(In reply to re: Some were discovered.... by Brian Smith)

I searched up to 2000 >= c >= b >=a >=1 and got only five more sets than Brian Smith: 92.

In each set, a never exceeds 25.

The stats are:

quotient  # of occurrences
 1             23 
 2             14
 3             13
 4             5
 5             12
 6             12
 8             7
 9             6

Interestingly, the first occurrences of the quotients appear in the order 9,8,6,5,4,3,2,1:

 1             1             1             9 
 1             1             2             8
 1             2             3             6
 1             4             5             5
 2             2             4             4
 2             4             6             3
 3             6             9             2
 5            20            25             1

even though 1 is the most common quotient.

Dim quot(100)

Private Sub Command1_Click()
 Open "natural numbers.txt" For Output As #2
 For a = 1 To 2000
  Text1.Text = Str(a)
 For b = a To 2000
 For c = b To 2000
  DoEvents
  q = (a + b + c) * (a + b + c) / (a * b * c)
  If q = Int(q) Then
    If quot(q) = 0 Then Print a, b, c, q
    quot(q) = quot(q) + 1
    Print #2, a, b, c, q
    Text2.Text = Str(a)
    Text3.Text = Str(b)
    Text4.Text = Str(c)
    Text5.Text = Str(q)
 
  End If
 Next
 Next
 Next
 For i = 1 To 100
  If quot(i) > 0 Then Print #2, i, quot(i)
 Next
 Close 2
 Print "done"
 
End Sub


  Posted by Charlie on 2007-10-15 14:19:53
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