A019280
Ralf Stephan
ralf at ark.in-berlin.de
Fri Jan 17 11:44:17 CET 2003
Jud McCranie:
> At 09:49 AM 1/16/2003 -0800, Marc LeBrun wrote:
> > >=Ralf Stephan
> > > the sequence is IMHO irrelevant, as is easily seen. Please comment
> > > if you think this is not so.
Make that 'should be deleted or renamed or edited'.
> ...
> Regarding A019280 specifically:
> >it's obviously been there a while,
Good one.
> >it has attracted extensions,
> >it is correct so far,
> >it can be extended to accommodate future odd superperfects (interpreting
> >"log_2" liberally), and
> >it will be non-trivial if odd superperfects exist (and useful if they are
> >too large for A019279).
>
> I agree with that. If another one is discovered, it will probably be too
> big for A19279, whereas it will fit in A19280.
Do you mean 'another even one' or 'odd'? The sequence could be filled up
the instant with even ones[*] and what you have is then 'Mersenne prime
exponent minus one with possible extension...' or you have 'log2 of etc'
with 'equal to Mersenne etc except possible...', and the adequate floor()
around log_2. Consequentially, the proof below would have to be added.
Really? What's the state of odd perfects?
ralf
[*] The reference contains the proof that any even superperfect is of the
form 2^(p-1) with 2^p-1 prime, but the reverse direction also holds since,
for all 2^k, sigma(2^k)=2^(k+1)-1, and, since that is prime for any Mersenne
prime, sigma(2^p-1)=2^p, so sigma(sigma(2^(p-1)))=2*2^(p-1) which is the
definition of superperfect. So every Mersenne exponent defines an even
superperfect. This might be at first have been overlooked.
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