[seqfan] Re: Self-stuffable numbers

[Ray Chandler]

John,
21021021021021021021 is a primitive root so 210210210210210210210 is self-stuffable and a member of A322323.  

But 21021021021021021021 also ends in a digit equal to the sum of digits minus number of digits, i.e. 21 - 20 = 1, and additionally 21021021021021021021 divides the result of self-stuffing so is also self-stuffable.  Therefore 21021021021021021021 is a member of A322323 and ends in something other than 0.  

See examples in A322002:
a(13) = 22 has E(22) = 4 - 2 = 2 = last digit of 22, so not only (2200, 220022002200, ...) but also (22, 2200220022, 220022002200220022, ...) is a subsequence of A322323. 
a(31) = 126 has E(126) = 9 - 3 = 6 = last digit of 126, so not only (126000000, 126000000126000000126000000, ...) but also (126, 126000000126000000126, ...) is a subsequence of A322323.

M | S(M) should be added to these examples.  We don't know yet what term of A322002 21021021021021021021 is but it is certainly a term.  

Now these are the only three known primitive roots that are themselves self-stuffable - 22, 126 and 21021021021021021021.   There may be others.  

Does this help any?  
Ray



> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Mason
> John
> Sent: Wednesday, January 2, 2019 2:05 PM
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Self-stuffable numbers
> 
> Ray
> I'm replying from my phone so maybe this reply won't be in the correct
> sequence.
> My question of terminating digit being 0,2,6 referred to self stuffable
> numbers, and not roots.
> So I think that 21 thing, which is a root and not a term of the main sequence,
> introduces no contradiction.
> The 21 number becomes a term only when terminating with 0, right?
> 
> Ottieni Outlook per Android<https://aka.ms/ghei36>
> 
> ________________________________
> From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of Ray Chandler
> <rayjchandler at sbcglobal.net>
> Sent: Wednesday, January 2, 2019 8:10:34 PM
> To: 'Sequence Fanatics Discussion list'
> Subject: [seqfan] Re: Self-stuffable numbers
> 
> A322323 has been restored to previous state.  Thanks for finding a couple of
> new primitive roots.
> Ray
> 
> > -----Original Message-----
> > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Lars
> > Blomberg
> > Sent: Wednesday, January 2, 2019 12:52 PM
> > To: 'Sequence Fanatics Discussion list' <seqfan at list.seqfan.eu>
> > Subject: [seqfan] Re: Self-stuffable numbers
> >
> > That is because my algorithm only finds values ending in 0, which of
> > course in an unwarranted assumption.
> > My last comment and b-file should be removed, but I am afraid I don't
> > know where to find the previous one.
> >
> > Sorry!
> > /Lars B
> >
> > -----Ursprungligt meddelande-----
> > Från: SeqFan <seqfan-bounces at list.seqfan.eu> För Ray Chandler
> > Skickat: den 2 januari 2019 19:29
> > Till: 'Sequence Fanatics Discussion list' <seqfan at list.seqfan.eu>
> > Ämne: [seqfan] Re: Self-stuffable numbers
> >
> > Unfortunately primitive root 21021021021021021021 adds another final
> > digit to your list.
> >
> > It is self-stuffable and therefore joins the ranks of 22 and 126 in
> > being primitive roots not ending in 0.
> >
> > Which begs the question why 21021021021021021021 is not in Lars'
> > b-file for A322323.
> > Ray
> >
> > >
> > > I agree.
> > >
> > > Btw, has anyone an idea how to prove the last digit is always 0,2,6?
> > >
> > > John
> > >
> > >
> > > >MFH: "The main challenge is to find larger (primitive) roots, i.e.,
> > > >terms of
> > > > A322323 (without repetitions and) with trailing zeros removed.
> > > >Arbitrarily  many larger terms of A322323 are then computed
> > > straightforwardly."
> > > >
> > > > Lars Blomberg has found 210210210210210210210 to be self-stuffable.
> > > > While it is trivially true that it could have been computed from
> > > > the primitive root 21021021021021021021, I don't believe that a
> > > > b-file for
> > > > A322002 with terms that large is in the offing.
> > >
> > > Ditto 52000005200000520000 with primitive root 5200000520000052 in Lars'
> > > new b-file for A322323.
> > > Ray
> > >
> > >
> > >
> > > --
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