🚀 go-pugleaf

Henrik, I tried to send this message to you privately but it bounced back
"user unknown".  If you see this please e-mail me at my address below.



From:	CPVA::SINGERM      "MARK SINGER <SINGERM@CPVA.SAIC.COM>" 22-JUL-1992 09:33:19.76
To:	SMTP%"Henrik.Johansson@nexus.comm.se"
CC:	SMTP%"Bumby@math.rutgers.edu",SINGERM
Subj:	sid(276)  and others

Hello Henrik,

I recently read a reply to a message of yours having to do with sums of
divisors, or, as you called it, the 'sid' function.  The response came
from R.T. Bumby, I never saw your original posting.  In any case, this
question intrigued me.  I am a mathematician who doesn't get much chance
nowadays to practice mathematics.  In particular, I enjoy anything that
relates to number theory and uncovering heretofore unknown patterns of
numbers.  (Sometime I'll tell you about the problem I've been working
on for the last 30 years.)

Enough small talk, now to the subject at hand:

>some numbers.  330 gave rise to this sequence

>...[data omitted]....

I have written a computer program to generate the sid sequences for any
seed number.  Following is my list of results for 330.  I'd appreciate
if you could match them against your list and confirm the accuracy of
my program for me:

330
534
546
798
1122
1470
2634
2646
4194
4932
7626
8502
9978
9990
17370
28026
35136
67226
33616
37808
40312
35288
37072
45264
79728
146448
281166
281178
363942
424638
526338
722961
321329
1

(No, I didn't type these, my program stores them in a VMS file which I
included here.)


>which also reached '1' eventually.  Then I tried 276 - this was a hard
>one!  The search is still not finished, because the numbers get bigger
>and bigger almost all the time!

Next, I tried 276 as the seed in my program.  The list *so far* follows:

276
396
696
1104
1872
3770
3790
3050
2716
2772
5964
10164
19628
19684
22876
26404
30044
33796
38780
54628
54684
111300
263676
465668
465724
465780
1026060
2325540
5335260
11738916
23117724
45956820
121129260
266485716
558454764
1092873236
1470806764
1471882804
1642613196
2737688884
2740114636
2791337780
4946860492
4946860548
9344070652
9344070708
15573451404
27078171764
27284104204
27410152084
27410152140
76787720100
220578719452
254903331620
361672366300
603062136740
921203207260
1381419996068
1395444575644
1395478688996
1395546402460
2069258468900
3065057872156
3277068463844
3429776547484
3597527970596
4028517592540
5641400009252

Since I can only use the computer for this purpose at night, I will start
it up again tonight using the seed 5641400009252 and continue the list.
I will do this every night until it either terminates or the number of
digits exceeds the limits of my system (around 10^20, I believe).


>Is there anybody that has a proof that any non-perfect number will,
>through repeated application of 'sid', give the number 1?  Except for
>numbers like 220 and 284 that gives the other.  Btw - is there
>sequences like sid(a) = b, sid(b) = c, sid(c) = d, sid(d) = a with a,
>b, c, and d all different?

My program doesn't *prove* anything, but it does verify the perfect
numbers 6, 28 and 496.  It also verifies the pair 220 and 284.  I have
yet to find another pair, or another longer sequence.  But I intend to
try.

Mr Bumby's response to you included the following paragraph:

>The sequence starting with 276 is mentioned as being not known to
>terminate in several of the reviews, but I don't know the current
>state of knowledge.  One of the computational papers tabulated
>sequences that start below 10^5 and eventually exceed 10^11.  

I expect to go beyond 10^11 in the next few days on the 276 problem,
as high as 10^20 or so if necessary, so maybe I'll be able to report
something to you later.


This is certainly an interesting problem, with several interesting
smaller problems:

1.   Are there other pairs such as 220/284?  Can they be generated?
2.   Are there similar triples, or longer sequences?
3.   Is there a relationship between the seed, the number of divisors
     of the seed and the length of the sequence required to reach 1.
4.   Is the distribution of perfect numbers random, or is there a
     pattern which can be used to generate additional perfect numbers?
        (I remember reading something about this a long time ago,
         but I'll have to try to find it again.  I think it had
         something to do with Mersenne primes.)



In conclusion, this message has not given you any answers, but I just
wanted to let you know that there is someone else interested enough in
the problem to give it some time and effort.  I intend to research the
references that Mr. Bumby gave you in his response, and see if I can
come up with anything.  Please keep me posted on any further progress
or discoveries you may make - I will certainly do the same for you.

Best regards,

Mark Singer
email:  SINGERM@CPVA.SAIC.COM

BTW, let me know if you are interested in the other problem I am working
on.  It relates to, but IS NOT an attempt to prove, Fermat's Last Theorem.
(Although a proof of Fermat's Last Theorem will be a straight forward
corollary of my work, if my work can ever be concluded.)  I welcome help
and other ideas as to how I can proceed, and would be delighted to send
you a separate message outlining the problem and the results I have
acheived to date.