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A Divisor Diversion (Posted on 2019-03-27) Difficulty: 3 of 5
Find the sum of all positive integers n, each of which has precisely 16 positive divisors and a successor n+1 that has precisely 3 divisors.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 5.0000 (1 votes)

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Some Thoughts computer exploration--probable solution | Comment 2 of 5 |
The only numbers with exactly three divisors are the squares of primes, so we need only check one less than the square of each prime to see if it has 16 divisors.

DefDbl A-Z
Dim fct(20, 1), crlf$
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
 
 Do
   p = nxtprm(p): pord = pord + 1
   n = p * p - 1
   f = factor(n)
   t = 1
   For i = 1 To f
     t = t * (fct(i, 1) + 1)
   Next
   If t = 16 Then
     tot = tot + n
     Text1.Text = Text1.Text & n & Str(p) & Str(p * p) & "    " & Str(tot) & crlf
   End If
   DoEvents
 Loop
 


Text1.Text = Text1.Text & tot & "done"
  
End Sub
Function factor(num)
 diffCt = 0: good = 1
 n = Abs(num): If n > 0 Then limit = Sqr(n) 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 n > 1 Then diffCt = diffCt + 1: fct(diffCt, 0) = n: fct(diffCt, 1) = 1
 factor = diffCt
 Exit Function

DivideIt:
 cnt = 0
 Do
  q = Int(n / dv)
  If q * dv = n And n > 0 Then
    n = q: cnt = cnt + 1: If n > 0 Then limit = Sqr(n) 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

Function prmdiv(num)
 Dim n, dv, q
 If num = 1 Then prmdiv = 1: Exit Function
 n = Abs(num): If n > 0 Then limit = Sqr(n) 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
 Loop
 If n > 1 Then prmdiv = n
 Exit Function

DivideIt:
 Do
  q = Int(n / dv)
  If q * dv = n And n > 0 Then
    prmdiv = dv: Exit Function
   Else
    Exit Do
  End If
 Loop

 Return
End Function

Function nxtprm(x)
  Dim n
  n = x + 1
  While prmdiv(n) < n
    n = n + 1
  Wend
  If n = 1 Then n = 2
  nxtprm = n
End Function

finds only

 n   p n+1    total so far
120 11 121     120
168 13 169     288

This is the only result so far when the program was checked at the 449526th prime, 6574639.

I'd say the answer is 288.

Edited on March 27, 2019, 5:22 pm
  Posted by Charlie on 2019-03-27 17:22:04

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