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?
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
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Posted by Charlie
on 2016-01-10 14:12:52 |
some stats
