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Sequence starting with 30 (Posted on 2016-07-05) Difficulty: 3 of 5
The prime factors of 30 are {2,3,5}
The sum of the reciprocals is 1/2+1/3+1/5=31/30
The product of the reciprocals is 1/2*1/3*1/5=1/30
The difference of the two is 31/30-1/30=1

There aren't many numbers where the final value of this procedure is a whole number.

Find more.

Other tasks:
Show that they cannot be semi-prime.
Show they must be square-free.

No Solution Yet Submitted by Jer    
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Some Thoughts program; research; program | Comment 1 of 3
   10   for N=2 to 35000
   20    Sum=0:Prod=1
   30    N2=N
   40    while N2>1
   50        F=prmdiv(N2)
   60        N2=N2//F
   70        Sum=Sum+1//F
   80        Prod=Prod//F
   90    wend
  100    Diff=Sum-Prod
  110    if Diff=int(Diff) and N<>prmdiv(N) then print N
  120   next N

finds only 30, 858 and 1722.

The "and N<>prmdiv(N)" is to weed out results that are prime, which derive from all prime n.

If these are the only three, then the "other tasks" follow immediately.

However, there are more, listed as Sloane's A007850.

Here are a few more from Sloane, factored into primes:

 30      2  3  5 
 858      2  3  11  13 
 1722      2  3  7  41 
 66198      2  3  11  17  59 
 2214408306      2  3  11  23  31  47057 
 24423128562      2  3  7  43  3041  4447 
 432749205173838      2  3  7  59  163  1381  775807 
 14737133470010574      2  3  7  71  103  67213  713863 
 550843391309130318      2  3  7  71  103  61559  29133437 
 244197000982499715087866346      2  3  11  23  31  47137    
   
 from

 
   10   open "a007850.txt" for input as #1
   15   open "giugax.txt" for output as #2
   20   while eof(1)=0
   30    input #1,G
   35    Giuga=val(G)
   40    print Giuga;"    ";
   41    print #2,Giuga;"    ";
   50    while Giuga>1
   60       Pf=prmdiv(Giuga):Giuga=Giuga//Pf
   70       print Pf;
   71       print #2,Pf;
   80    wend
   85   print
   86   print #2,
   90   wend
  100   close #2

Sloane had a couple more but they exceeded the limitations of UBASIC's prmdiv function.

  Posted by Charlie on 2016-07-05 14:43:03
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