[seqfan] Re: number theory question about A000593

[Jack Brennen]

Your second paragraph is equivalent to:

   sigma(m) == k*m+1,  k > 1

If k == 2, this is the question of the existence of quasiperfect numbers.
It's been published that no quasiperfect number exists < 10^35.  I've
verified myself that no quasiperfect number exists < 10^80, and I'm
currently working on extending that to 10^120.  Any quasiperfect number
would have to be an odd square, in addition to a bunch of other
restrictions.  Heuristically, there are unlikely to be any such numbers.

I think it's more likely to find solutions with k == 3, but any such
solutions wouldn't be relevant to Peter's question.  Such numbers should
be very scarce, mainly because both m and sigma(m) are likely to have
many divisors, and yet they must be coprime.

    Jack

On 6/18/2011 1:18 PM, Max Alekseyev wrote:
> Basically this conjecture is equivalent to whether there exists an odd
> number m such that (sigma(m)-1)/m equals 2^k for some integer k>0.
>
> I do not know even if there exists an integer m such that
> (sigma(m)-1)/m is integer greater than 1 (equivalently, sigma(m) == 1
> (mod m) and sigma(m)>m+1).
>
> Max
>
> On Sat, Jun 18, 2011 at 7:37 PM, Peter Lawrence
> <peterl95124 at sbcglobal.net>  wrote:
>>
>> I was tinkering with A000593 "sum of odd divisors of N" and noticed that
>> empirically
>>
>> A(n)  =  n+1    IFF    isprime(n)<--- conjecture ???
>>
>>
>> (a bit of finagling results in only needing to check for odd squares, so a
>> simple C program
>> verifies this out to 10^12 pretty fast)
>>
>>
>> So I am wondering if this is actually true, and if so is there is any N.T.
>> proof.
>>
>>
>>
>> apologies in advance if this is too off-topic for this list,
>> thanks,
>> Peter Lawrence.
>>
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