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 Sequence starting with 30 (Posted on 2016-07-05)
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.

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

 No Solution Yet Submitted by Jer No Rating

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 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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