[seqfan] Re: Stumped on sums of perfect numbers

[franktaw at netscape.net]

I think you get this result if you say "multiply perfect" instead of 
"perfect" (but excluding 1), and mistakenly include sigma(672) = 2016 
instead of 672 itself. (The reference is to A065997, which is n such 
that sigma(n)/n is prime; this may be what is intended. 30240 is the 
first difference from what I have suggested, and these values do not 
get that high.)

Franklin T. Adams-Watters

-----Original Message-----
From: Alonso Del Arte <alonso.delarte at gmail.com>

Yesterday, Harvey noticed that A083865 doesn't quite fit with its
definition, "perfect numbers and sums of perfect numbers." I am
convinced that there is some thought process behind that sequence
which wasn't written down and may now very well be forgotten. I have
e-mailed Torsten, but while I wait for a response, I have pondered a
number of different explanations, none of which are satisfactory for
the numbers given.

It is possible that perfect numbers may be used more than once. Thus,
120 = 3(28) + 6^2. But then why are numbers like 112 and 118 not in
the sequence?

Or maybe perfect numbers may be "repeated" if they have already been
added in. Since 6 + 28 = 34, under this explanation we could then
(after a(3)) do 6 + 28 + 34 = 68. But 68 is not in the sequence.

Maybe k-perfect numbers are allowed. That would explain the presence
of 120, as it is 3-perfect. But then why is 672 missing? And likewise
678, 700, 792, etc.

I think that Harvey's interpretation of the current definition 6, 28,
34, 496, 502, 524, 530, 8128, ... merits to be a new sequence in its
own right. And if we can't find a satisfactory explanation for
A083865, maybe it should become a dead sequence.

Al

--
Alonso del Arte
Author at 
SmashWords.com<https://www.smashwords.com/profile/view/AlonsoDelarte>
Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>

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