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Smooth Sum Situation (Posted on 2019-01-23) Difficulty: 4 of 5
OIES sequence A002473 is the sequence of 7-smooth numbers: positive numbers whose prime divisors are all less than or equal to 7.

Let S be the infinite summation of all the reciprocals of the members of A002473. Does the sum converge, and if so then what is the sum?

No Solution Yet Submitted by Brian Smith    
Rating: 3.0000 (1 votes)

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Some Thoughts computer sum; dumb way and smarter way | Comment 1 of 3
While the computer calculation of the sum is not a proof that the series converges, it does seem apparent that it does: to 35/8.

I started with a dumb way of getting the sum, and then the smarter way.

First the dumb way: examine all the numbers up to 100 million and add in the reciprocals of only those numbers that have no higher prime factor than 7:

10000000 4.37494579735816
20000000 4.3749699085652
30000000 4.37497872059798
40000000 4.37498334970393
50000000 4.374986284036
60000000 4.37498825720054
70000000 4.37498973931217
80000000 4.37499084676966
90000000 4.37499174203155
100000000 4.37499243708591

It looks like it might be approaching 35/8. But the program wasted a lot of time factoring all the numbers up to 100 million:

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

Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function

Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 
 For n = 1 To 100000000
  f = factor(n)
  If fct(f, 0) <= 7 Then tot = tot + 1 / n
  If n Mod 10000000 = 0 Then Text1.Text = Text1.Text & n & Str(tot) & crlf
  DoEvents
 Next
 
 Text1.Text = Text1.Text & Str(tot2)
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

The smarter way is to vary the powers of 2, 3, 5 and 7. Only usable numbers are used, so we can include some numbers much larger than 100 million, but also take less time:

 4.37499996661196
 
 It's looking even better for 35/8.
 
 DefDbl A-Z
 Dim crlf$, fct(20, 1)
 
 Function mform$(x, t$)
   a$ = Format$(x, t$)
   If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
   mform$ = a$
 End Function
 
 Private Sub Form_Load()
  Text1.Text = ""
  crlf$ = Chr(13) + Chr(10)
  Form1.Visible = True
  
  pwr2 = 1
  
  For p2 = 0 To 27
  pwr3 = 1
  For p3 = 0 To 17
  pwr5 = 1
  For p5 = 0 To 12
  pwr7 = 1
  For p7 = 0 To 10
    tot = tot + 1 / (pwr2 * pwr3 * pwr5 * pwr7)
    pwr7 = pwr7 * 7
  Next
    pwr5 = pwr5 * 5
  Next
    pwr3 = pwr3 * 3
  Next
    pwr2 = pwr2 * 2
  Next
  
  Text1.Text = Text1.Text & Str(tot)
 End Sub

Further modification:

 4.37499999751474
 
 better still for 35/8, using these limits on the powers:
 
  For p2 = 0 To 30
  pwr3 = 1
  For p3 = 0 To 20
  pwr5 = 1
  For p5 = 0 To 15
  pwr7 = 1
  For p7 = 0 To 15
  
But why not go as high as will affect the total?  In fact, if the powers are allowed to go so high as to have no effect on the total we get  4.37499999999873.

code fragment:

 pwr2 = 1
 
 For p2 = 0 To 300
 pwr3 = 1
 For p3 = 0 To 300
 pwr5 = 1
 For p5 = 0 To 300
 pwr7 = 1
 For p7 = 0 To 300
   tot = tot + 1 / (pwr2 * pwr3 * pwr5 * pwr7)
   pwr7 = pwr7 * 7
   If pwr7 > 100000000000000# Then Exit For
 Next
   pwr5 = pwr5 * 5
   If pwr5 > 100000000000000# Then Exit For
 Next
   pwr3 = pwr3 * 3
   If pwr3 > 100000000000000# Then Exit For
 Next
   pwr2 = pwr2 * 2
   If pwr2 > 100000000000000# Then Exit For
 Next
 
 Text1.Text = Text1.Text & Str(tot)


  Posted by Charlie on 2019-01-23 15:42:55
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