[seqfan] Re: Nice new sequence with interesting open problem
Neil Sloane
njasloane at gmail.com
Fri Sep 13 14:03:55 CEST 2019
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Andrew, That's great! I see that Remy Sigrist already added a proof that
the sequence is bounded.
But I think we should have your proof too. Could you add your proof as a
plain text file attached
to the entry? It will end up with a file name like a326344_1.txt,
I mentioned this sequence at the Rutgers Experimental Math seminar
yesterday, and some of the students got interested right away, so please do
add your proof to the entry (they won't have seen it, not being on the
mailing list).
Best regards
Neil
Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com
On Fri, Sep 13, 2019 at 4:51 AM Andrew Weimholt <andrew.weimholt at gmail.com>
wrote:
> Hi Neil,
>
> Here's a proof that 909 is the highest number reached...
>
> Each number in the sequence has two possible successors, so if we just look
> at
> what values can succeed each other without regard to their position in the
> sequence, n,
> we get a binary tree.
>
> However, since n can only be prime when it is congruent to +/-1 mod 6, some
> otherwise potential successors can never be reached.
>
> The transition from single digits, to double digits always occurs with
> a(n)=11,
> and n can either be congruent to -1 or 1 mod 6, so we get the following two
> trees,
> shown below and continuing until the branches transition either to 3 digits
> or back to 1 digit...
>
> n mod 6
>
> —————————————————————————————————————————
>
> 1 11
>
> 2 21
>
> 3 22
>
> 4 42_______________________
>
> 5* 44 34
>
> 0 54___________ 53______
>
> 1* 55 95 45 95
>
> 2 65 69 64 69
>
> 3 66 [7] 56 [7]
>
> 4 86______ 75___________
>
> 5* 78 98 67 97
>
> 0 [8] 99__________ 86______ 89______
>
> 1* [1] [101] 78 98 [9] 79
>
> 2 [8] 99 [8]
>
> 3 [1]
>
>
>
> n mod 6
>
> —————————————————————————————————————————
>
> 5 11
>
> 0 21__________________________
>
> 1* 22 32
>
> 2 42 33
>
> 3 44 43
>
> 4 54____________ 44__________________
>
> 5* 55 95 54 74
>
> 0 65______ 69_______ 55______ 57____________
>
> 1* 66 76 [7] 17 65 95 85 95
>
> 2 86 77 81 66 69 68 69
>
> 3 78 87 28 86 [7] 96 [7]
>
> 4 [8] 88______ [3] 78______ 89______
>
> 5* [9] 98 [8] 97 [9] 79
>
> 0 99________ 89______ [8]
>
> 1* [1] [101] [9] 79
>
> 2 [8]
>
> Both of these trees can transition to 101, but 101 always occurs when n is
> congruent to 1 mod 6,
> so we only get 1 tree of three digits numbers for 101.
>
> n mod 6
>
> —————————————————————————————————————————
>
> 1 101
>
> 2 201
>
> 3 202
>
> 4 302_______________
>
> 5* 303 703
>
> 0 403_______ 407_______
>
> 1* 404 904 804 904
>
> 2 504 509 508 509
>
> 3 505 [15] [15] [15]
>
> 4 605_______________
>
> 5* 606 706
>
> 0 806_______ 707_______
>
> 1* 708 908 807 907
>
> 2 [17] 909 808 809
>
> 3 [19] [18] [18]
>
>
> Now we are left with four cases for transitioning back to
> 2 digits from 3 digits...
>
> 18 and 19 will immediately transition all the way back to
> 1 digit, as the next position will be congruent to 4 mod 6,
>
> and the in both cases, the next composite is 20, which backwards is 2.
>
> 17 gives us the tree...
>
> n mod 6
>
> —————————————————————————————————————————
>
> 2 17
>
> 3 81
>
> 4 28______
>
> 5* [3] 92
>
> 0 39______
>
> 1* [4] 14
>
> 2 51
>
> 3 25
>
> 4 62____________
>
> 5* 36 76
>
> 0 83______ 77______
>
> 1* 48 98 87 97
>
> 2 94 99 88 89
>
> 3 59 [1] [9] [9]
>
> 4 [6]
>
>
> and 15 gives us...
>
>
> n mod 6
>
> —————————————————————————————————————————
>
> 3 15
>
> 4 61___________________
>
> 5* 26 76
>
> 0 72______ 77______
>
> 1* 47 37 87 97
>
> 2 84 83 88 89
>
> 3 58 48 [9] [9]
>
> 4 [6] 94______
>
> 5* 59 79
>
> 0 [6] [8]
>
>
> Therefore the transitions into 3 digit are always followed very soon
> by transitions back to 2 digits and soon after back to 1 digit.
>
> Andrew
>
>
>
>
> On Thu, Sep 12, 2019 at 10:20 AM Neil Sloane <njasloane at gmail.com> wrote:
>
> > Sent in by Max Tohline
> >
> > A326344 a(1) = 1. Thereafter, if n is prime, a(n) is the next prime after
> > a(n-1), but written backwards. If n is not prime, a(n) is the next
> > composite after a(n-1), written backwards.
> >
> >
> > 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 4, 5, 6, 8, 9, 11, 21, 32, 33, 43, 44,
> 74,
> > 57, 85, 68, 96, 89, 79, 8, 11, 21, 22, 42, 44, 54, 95, 69, 7, 8, 11, 21,
> > 32, 33, 43, 44, 74, 57, 85, 68, 96, 89, 79, 8, 9, 1, 4, 6, 7, 8, 11, 21,
> > 22, 42, 44, 54, 95, 69, 7, 8, 11, 21, ...
> >
> >
> > It appears that this does not exceed 909, reached for the first time at
> n =
> > 21752
> >
> > Michel Marcus checked it up to 10^8.
> >
> > It would be nice to know more!
> >
> > The record high values are:
> >
> > 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 21, 32, 33, 43, 44, 74, 85, 96, 97, 98,
> 99,
> > 101, 201, 202, 302, 703, 804, 806, 807, 808, 907, 908, 909
> > (as far as they are known, see A326298, A326402)
> >
> > Neil
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
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