Every composite number N can be represented by a product of k (k>1) primes, not necessarily distinct,
N=p1, p2, p3, … plast, e.g.
28=2*2*7.
In this case, the sum of al primes from the smallest to the last, including both is
2+3+5+7=17, which differs from the number
28.
What is the smallest composite number that equals the inclusive sum of the primes from its smallest to the largest prime factor, including both ?
How many additional numbers, displaying the said feature exist below 200?
The smallest four are listed in the first column below. The prime addends are shown to the right.
10 2 3 5
39 3 5 7 11 13
155 5 7 11 13 17 19 23 29 31
371 7 11 13 17 19 23 29 31 37 41 43 47 53
Looking up the sequence of numbers in Sloane's OEIS finds A055233, indicating that the next two in this sequence are 2935561623745 and 454539357304421.
The same four numbers also begin the sequence A055514, but that sequence has a less stringent criterion for membership: that the sum of the consecutive primes is divisible by the first and last primes, perhaps also having a smaller divisor as well. not requiring that the primes be summed back to that prime.
DefDbl A-Z
Dim crlf$, fct(20, 1)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
For n = 4 To 999
DoEvents
f = factor(n)
p = fct(1, 0): lastp = fct(f, 0)
p1 = p
s = 0
Do
s = s + p
p = nxtprm(p)
Loop Until p > lastp
If s = n And f > 1 Then
Text1.Text = Text1.Text & n & " "
p = p1
Do
Text1.Text = Text1.Text & Str(p)
p = nxtprm(p)
Loop Until p > lastp
Text1.Text = Text1.Text & crlf
End If
Next
n = 2935561623745#
Text1.Text = Text1.Text & crlf & " done"
End Sub
Function factor(num)
diffCt = 0: good = 1
n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0
If limit <> Int(limit) Then limit = Int(limit + 1)
dv = 2: GoSub DivideIt
dv = 3: GoSub DivideIt
dv = 5: GoSub DivideIt
dv = 7
Do Until dv > limit
GoSub DivideIt: dv = dv + 4 '11
GoSub DivideIt: dv = dv + 2 '13
GoSub DivideIt: dv = dv + 4 '17
GoSub DivideIt: dv = dv + 2 '19
GoSub DivideIt: dv = dv + 4 '23
GoSub DivideIt: dv = dv + 6 '29
GoSub DivideIt: dv = dv + 2 '31
GoSub DivideIt: dv = dv + 6 '37
If INKEY$ = 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
nxtprm = n
End Function
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Posted by Charlie
on 2016-04-04 11:21:07 |
computer solution
