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A series of primes equals N (Posted on 2018-10-19) Difficulty: 3 of 5
N=sum of primes between smallest and largest prime factor of N (inclusive).

Find all possible values for the composite number N below 300.

No Solution Yet Submitted by Ady TZIDON    
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Solution computer solution Comment 2 of 2 |
It's interesting that the cutoff was made at 300, when going just a bit farther, to 371, gives the last example at least as far out as 30,000.  I don't know if 371 is the last in a finite set or just that beyond this number the examples are extremely rarified. I haven't found a trend toward either the total or the number itself as being the larger, so that we could say that beyond a certain point one is always larger than the other.

10   2-5
2 * 5
2 + 3 + 5

39   3-13
3 * 13
3 + 5 + 7 + 11 + 13

155   5-31
5 * 31
5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31

371   7-53
7 * 53
7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53

After finding these, a search in the OEIS finds A055233, where the next two are given as 2935561623745 (5 through 9557887) and  454539357304421 (3536123 through  128541727). This is not to be confused with the basis for A055514, which does not require that the sequence start with the lowest prime factor of n (nor necessarily end with the highest, for that matter).

DefDbl A-Z
Dim crlf$, fct(20, 1)


Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 
 Form1.Visible = True
 
 For n = 6 To 30000
   tot = 0: lim = Sqr(n): n1 = n
   firstPrime = 0
   st = 0
   pNo = 0
   Do
     pNo = pNo + 1: p = prm(pNo)
     If p > lim Then Exit Do
     q = Int(n1 / p): r = n1 - q * p
     If r = 0 Then st = 1: If firstPrime = 0 Then firstPrime = p
     If st Then tot = tot + p
     While r = 0
       n1 = n1 / p
       q = Int(n1 / p): r = n1 - q * p
     Wend
     lim = Sqr(n1)
     DoEvents
   Loop
   If n1 > 0 And firstPrime > 0 Then
     For p1 = pNo To 10000
       If prm(p1) > n1 Then Exit For
       tot = tot + prm(p1)
     Next
   End If
   If tot = n Then
     Text1.Text = Text1.Text & n & "   " & firstPrime & "-" & prm(p1 - 1) & crlf
     f = factor(n)
     For i = 1 To f
       Text1.Text = Text1.Text & fct(i, 0)
       If fct(i, 1) > 1 Then Text1.Text = Text1.Text & "^" & fct(i, 1)
       If i < f Then Text1.Text = Text1.Text & " * "
     Next
     Text1.Text = Text1.Text & crlf
     p = firstPrime
     Do
       Text1.Text = Text1.Text & p
       If p = prm(p1 - 1) Then Exit Do
       Text1.Text = Text1.Text & " + "
       p = nxtprm(p)
     Loop
     Text1.Text = Text1.Text & crlf & crlf
   End If
   DoEvents
 Next n
  
 Text1.Text = Text1.Text & crlf & "done"
  
End Sub

Function prm(i)
  Dim p As Long
  Open "17-bit primes.bin" For Random As #111 Len = 4
  Get #111, i, p
  prm = p
  Close 111
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 Or n < 2
    n = n + 1
  Wend
  nxtprm = n
End Function

 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


  Posted by Charlie on 2018-10-19 11:53:38
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