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Arithmetic Product Puzzle (Posted on 2015-08-13) Difficulty: 3 of 5
Determine the smallest positive integer that is expressible as the product of three distinct positive integers in arithmetic sequence in precisely two ways.

What are the next two smallest positive integers with this property?

**** As an example, 105 is expressible as the product of three positive integers (3, 5 and 7) in arithmetic sequence in only one way as no other positive integer triplet in arithmetic sequence multiplies to 105.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution Comment 1 of 1

The first 7 (under 2000) are:

  n  ways
 231 2
 440 2
 504 2
 840 2
1560 2
1680 2
1848 2

The first three's details are:

 n  sequence
231 1 11 21
231 3 7 11
440 2 11 20
440 5 8 11
504 4 9 14
504 7 8 9

By rearranging the output of:

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)

 high = 2000
 ReDim ways(high)
 For a = 1 To high
 For d = 1 To high - a
   s = a * (a + d) * (a + 2 * d)
   If s <= high Then ways(s) = ways(s) + 1 Else Exit For
   If s = 231 Or s = 440 Or s = 504 Then
     Text1.Text = Text1.Text & s & Str(a) & Str(a + d) & Str(a + 2 * d) & crlf
   End If
 Next
 Next
 
 For i = 1 To high
   If ways(i) = 2 Then Text1.Text = Text1.Text & i & Str(ways(i)) & crlf: DoEvents
 Next
 
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub


The lines

   If s = 231 Or s = 440 Or s = 504 Then
     Text1.Text = Text1.Text & s & Str(a) & Str(a + d) & Str(a + 2 * d) & crlf
   End If

were obviously added after the first run determined the first three solutions.

  Posted by Charlie on 2015-08-13 14:52:35
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