[seqfan] Re: Stumped on sums of perfect numbers
Jonathan Post
jvospost3 at gmail.com
Thu Feb 9 02:18:35 CET 2012
One logical supersequence would be:
Sums of exactly two distinct perfect numbers:
34, 502, 524, 8134, 8156, 8624, 33550342, 8589877184, 8623419392,
137438691824, 8589869084, 8589869084, 8589877184, 137438691824,
137438691334, 137438691356, 137438691824, 137438699456,
2305843008139952134, 2305843008139952156, 2305843008139 952624,
33550336, 8589869056, 137438691328, 137472241664, 2305843008139952128,
2305843008139960256, 2305843008173502464
Sums of exactly three distinct perfect numbers:
530, 8652, ...
Sums of exactly four distinct perfect numbers:
8658, ...
and the supersequence with the same definition but without "distinct"
sums of exactly two (not necessarily distinct) perfect numbers:
12, 34, 56, 502, 524, 8134, 8156, 33550342, 33550364, 8589869084,
137438691334, 137438691356, 2305843008139952134, 2305843008139952156
sums of exactly three (not necessarily distinct) perfect numbers:
18, 40, 62, 508, 530, ...
sums of exactly four (not necessarily distinct) perfect numbers:
24, 46, 68, 90, 514, ...
Ooops: all of the above may be wrong due to sticky keyboard. But the
supersequence of definitions is still suggested. Better to program it,
rather than do it by hand, cutting and pasting from a calculator.
On Wed, Feb 8, 2012 at 4:24 PM, Alonso Del Arte
<alonso.delarte at gmail.com> wrote:
> 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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