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Primevens and primeodds (Posted on 2016-01-10) Difficulty: 4 of 5
Let a primeven be a positive integer that is the product of an even number of primes. Let a primeodd be a positive integer that is the product of an odd number of primes. Then, 1 is a primeven because it is the product of 0 primes. 2 is a primeodd because it is the product of 1 prime. 3 is a primeodd because it is the product of 1 prime. 4 is a primeven because it is the product of 2 primes. Here are the first 10 positive integers.
Number:   Factorization:      Number of primes:   Type:
1                             0                   primeven
2         2                   1                   primeodd
3         3                   1                   primeodd
4         2*2                 2                   primeven
5         5                   1                   primeodd
6         2*3                 2                   primeven
7         7                   1                   primeodd
8         2*2*2               3                   primeodd
9         3*3                 2                   primeven
10        2*5                 2                   primeven
Suppose the primevens and primeodds had a race. First, the primevens would be ahead because 1 is a primeven. Then, there would be a tie because 2 is a primeodd. Then, the primeodds would be ahead because 3 is a primeodd. Then, there would be a tie because 4 is a primeven. Here are the winners from 1 to 10.
Number:   Type:          Primevens:     Primeodds:     Winner:
1         primeven       1              0              primevens
2         primeodd       1              1              tie
3         primeodd       1              2              primeodds
4         primeven       2              2              tie
5         primeodd       2              3              primeodds
6         primeven       3              3              tie
7         primeodd       3              4              primeodds
8         primeodd       3              5              primeodds
9         primeven       4              5              primeodds
10        primeven       5              5              tie
The primevens were ahead at the start, but have not been ahead since then. Do the primevens ever become the winner again?

No Solution Yet Submitted by Math Man    
Rating: 4.0000 (1 votes)

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Some Thoughts some stats | Comment 2 of 8 |
In the first 10 million numbers, 1 is the only case where it happens.  However the numbers are close enough to mimic a random walk (difference is within the square root of the total).

1 1 0***

then totals shown every million:

   max     evens   odds
  number
 1000000   499735  500265
 2000000   999383 1000617
 3000000  1499529 1500471
 4000000  1999451 2000549
 5000000  2498854 2501146
 6000000  2999254 3000746
 7000000  3499209 3500791
 8000000  3998778 4001222
 9000000  4498714 4501286
10000000  4999579 5000421



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


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For i = 1 To 10000000
   DoEvents
   f = factor(i)
   pf = 0
   For j = 1 To f
     pf = pf + fct(j, 1)
   Next
   If pf Mod 2 = 0 Then evenct = evenct + 1 Else oddct = oddct + 1
   If i Mod 1000000 = 0 Then
     Text1.Text = Text1.Text & i & Str(evenct) & Str(oddct) & crlf
   End If
   If evenct > oddct Then
     Text1.Text = Text1.Text & i & Str(evenct) & Str(oddct) & "***" & crlf
   End If
 Next
 

 Text1.Text = Text1.Text & crlf & " done"
  
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


  Posted by Charlie on 2016-01-10 14:12:52
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