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SUM = PR0DUCT (Posted on 2017-10-18) Difficulty: 3 of 5
List all the integers N, below 10000, such that the sum of their proper divisors (i.e. N excluded) equals their product.

Counterexample:
take 12: s(1,2,3,4,6)=16; p(1,2,3,4,6)=144;
so 12 is not on the list.

See The Solution Submitted by Ady TZIDON    
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Solution computer solution | Comment 1 of 3
DefDbl A-Z
Dim fct(20, 1), crlf$, fsum, fprod, propDiv, f, n
Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function

Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 
 Form1.Visible = True
 
For n = 2 To 10000
  f = factor(n)
  fsum = 0: fprod = 1
  propDiv = 1
  doPropDiv 1
  DoEvents
  
  If fsum = fprod And fsum <> 1 Then
        Text1.Text = Text1.Text & Str(n) & Str(fsum) & crlf$
        DoEvents
  End If
Next n
End Sub

Sub doPropDiv(wh)
  For i = 0 To fct(wh, 1)
     pdSave = propDiv
     propDiv = Int(propDiv * fct(wh, 0) ^ i + 0.5)
     If wh < f Then
        doPropDiv wh + 1
     Else
        If propDiv < n Then
            fsum = fsum + propDiv
            fprod = fprod * propDiv
        End If
     End If
     propDiv = pdSave
  Next
End Sub

Function factor(num)
 diffCt = 0: good = 1
 nm1 = Abs(num): If nm1 > 0 Then limit = Sqr(nm1) Else limit = 0
 If limit <> Int(limit) Then limit = Int(limit + 1)
 dv = 2: GoSub DivideIt
 dv = 3: GoSub DivideIt
 dv = 5: GoSub DivideIt
 dv = 7
 Do Until dv > limit
   GoSub DivideIt: dv = dv + 4 '11
   GoSub DivideIt: dv = dv + 2 '13
   GoSub DivideIt: dv = dv + 4 '17
   GoSub DivideIt: dv = dv + 2 '19
   GoSub DivideIt: dv = dv + 4 '23
   GoSub DivideIt: dv = dv + 6 '29
   GoSub DivideIt: dv = dv + 2 '31
   GoSub DivideIt: dv = dv + 6 '37
   If INKEY$ = Chr$(27) Then s$ = Chr$(27): Exit Function
 Loop
 If nm1 > 1 Then diffCt = diffCt + 1: fct(diffCt, 0) = nm1: fct(diffCt, 1) = 1
 factor = diffCt
 Exit Function

DivideIt:
 cnt = 0
 Do
  q = Int(nm1 / dv)
  If q * dv = nm1 And nm1 > 0 Then
    nm1 = q: cnt = cnt + 1: If nm1 > 0 Then limit = Sqr(nm1) Else limit = 0
    If limit <> Int(limit) Then limit = Int(limit + 1)
   Else
    Exit Do
  End If
 Loop
 If cnt > 0 Then
   diffCt = diffCt + 1
   fct(diffCt, 0) = dv
   fct(diffCt, 1) = cnt
 End If
 Return
End Function

finds only 6 due to the additional requirement that the sum (and therefore product) fsum <> 1. Before that requirement was imposed, every prime number was also included, as it should be for the actual wording of the puzzle.

  Posted by Charlie on 2017-10-18 10:52:00
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