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 Special Natural Numbers (Posted on 2007-10-14)
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 No Rating

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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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