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Some questions of abundance (Posted on 2015-10-28) Difficulty: 2 of 5
Under what conditions is the number 2n*p an abundant number? Where n is a positive integer and p is prime.

How about 3n*p?

How about 2*pn?

Recall an abundant number is one whose proper factors sum to more than the number. For example 18 is abundant because 1+2+3+6+9 > 18.

No Solution Yet Submitted by Jer    
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re(2): proposed solution / counterexample Comment 3 of 3 |
(In reply to re: proposed solution / counterexample by Jer)

Doing some calculations via program I see


when n=1, none are abundant
n=2 highest prime giving abundance is 5
n=3 highest prime giving abundance is 13
n=4 highest prime giving abundance is 29
n=5 highest prime giving abundance is 61
n=6 highest prime giving abundance is 113
n=7 highest prime giving abundance is 251
n=8 highest prime giving abundance is 509
n=9 highest prime giving abundance is 1021
n=10 highest prime giving abundance is 2039
n=11 highest prime giving abundance is 4093
n=12 highest prime giving abundance is 8179
n=13 highest prime giving abundance is 16381
n=14 highest prime giving abundance is 32749
n=15 highest prime giving abundance is 65521

Except for n=1, all primes p less than 2^(n+1) - 1 work.

If 2^(n+1) - 1 is prime then that is the first non-working p and it results in a perfect number.

DefDbl A-Z
Dim crlf$


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

 For n = 2 To 15
  p = 1
  Do
   DoEvents
   ppp = pp
   pp = p
   p = nxtprm(p)
   If (p + 1) * ((2 ^ (n + 1)) - 1) - p * 2 ^ n > p * 2 ^ n Then sv = p
 '  Text1.Text = Text1.Text & n & Str(p) & "    "
 '  Text1.Text = Text1.Text & p * 2 ^ n & Str((p + 1) * ((2 ^ (n + 1)) - 1) - p * 2 ^ n) & crlf
  Loop Until (p + 1) * ((2 ^ (n + 1)) - 1) - p * 2 ^ n < p * 2 ^ n: Text1.Text = Text1.Text & crlf
  p = sv
  Text1.Text = Text1.Text & n & Str(p) & "    "
  Text1.Text = Text1.Text & p * 2 ^ n & Str((p + 1) * ((2 ^ (n + 1)) - 1) - p * 2 ^ n) & crlf
 Next
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

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
  nxtprm = n
End Function


  Posted by Charlie on 2015-10-28 13:06:43
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