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Topic: Solitary numbers / Friendly numbers (help needed)
Replies: 3   Last Post: Sep 17, 2004 4:30 PM

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David Einstein

Posts: 389
Registered: 12/6/04
Re: Solitary numbers / Friendly numbers (help needed)
Posted: Sep 15, 2004 9:11 AM
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Geert van der Wulp wrote:
> Lately I have taken up an interest in solitary numbers: numbers which
> do not have "friends". The number n has a friend m iff:
> sigma(n)/n = sigma(m)/m
> Now I have read and verified the claims that a number n has NO friends
> (so it is solitary) if:
> 1) n is prime,
> 2) n is a prime power,
> 3) if the greatest common divisor of n and sigma(n) is 1.
> Now another source claimed that for certain numbers it is not
> difficult to prove that they are solitary, even though they fall
> outside the above mentioned rules. This is supposedly the case for the
> number 18.
> Now I have tried to prove this claim, but I did not succeed:
> Assume first that we have found a number x for which
> sigma(x)/x = sigma(18)/18 = (1+2+3+6+9+18)/18 = 13/6
> This implies that 6 is a divisor of x and 13 is a divisor of sigma(x).
> How to proceed?

We know that 2*3=6 divides x, so we know that x=2^a * 3^b * n where n is
relatively prime to 6. Now it is easy to see that if sigma(x)/x = 13/6
and x != 18 then a=b=1. (Can you see why?) Now if x=6*n with n
relatively prime to 6 then the numerator of sigma(x)/x reduced to lowest
terms will be even, and the denominator will be relatively prime to 6,
in particular it will not equal 13/6.

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