[seqfan] Sharing no digit business (with building process)

[Eric Angelini]

> Please explain exactly how you determine the value of a(n).

... yes, sorry again to be unclear:

start with a(1) = 2
S = 2, ...
this "2" says: "the 2nd term of S shares no digit with me"
As we always extend S with the smallest integer not used so far
and not leading to a contradiction, we put "1"; now S is:

S = 2, 1, ...
this "1" says: "the 1st term of S shares no digit with me"
which is true.

Can we extend S with "3"? No -- this "3" would lie:

S = 2, 1, 3, ...
this "3" says: "the 3rd term of S shares no digit with me"
-- this is obviously false.

So we see that we quickly have for S:

S = 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, ...

Now the 11th term cannot be 11 -- as 11 shares at least one
"1" with 11 itself -- look (n = is the rank):

S = 2, 1, 4, 3, 6, 5, 8, 7, 10,  9, 11
n = 1  2  3  4  5  6  7  8   9  10  11

We then try 12 -- but again, 12 shares a "1" with n = 11:

S = 2, 1, 4, 3, 6, 5, 8, 7, 10,  9, 12
n = 1  2  3  4  5  6  7  8   9  10  11

So we have to jump to 20 and try -- 20 is ok:

S = 2, 1, 4, 3, 6, 5, 8, 7, 10,  9, 20
n = 1  2  3  4  5  6  7  8   9  10  11

What about the next term?

S = 2, 1, 4, 3, 6, 5, 8, 7, 10,  9, 20, ...
n = 1  2  3  4  5  6  7  8   9  10  11  12

This next term cannot contain any "1" or "2"; 30 fits:

S = 2, 1, 4, 3, 6, 5, 8, 7, 10,  9, 20, 30
n = 1  2  3  4  5  6  7  8   9  10  11  12

What about the next term?

S = 2, 1, 4, 3, 6, 5, 8, 7, 10,  9, 20, 30
n = 1  2  3  4  5  6  7  8   9  10  11  12  13

The smallest available integer containing no "1" and no "3"
is 22:

S = 2, 1, 4, 3, 6, 5, 8, 7, 10,  9, 20, 30, 22
n = 1  2  3  4  5  6  7  8   9  10  11  12  13

etc.

You are right, my first try (in my first msg), was bugged:

> S = 2,1,4,3,6,5,8,7,10,9,12,30,14,20,16,22,18,23,21,31,33,34,...
> (a nightmare to do by hand)

... this last sentence is not bugged at all, geee :-(

Hope this is sound now,
Best,
E.





-----Message d'origine-----
De : seqfan-bounces at list.seqfan.eu [mailto:seqfan-bounces at list.seqfan.eu] De la part de David Wilson
Envoyé : vendredi 3 avril 2009 19:10
À : Sequence Fanatics Discussion list
Objet : [seqfan] Re: Sharing no digit business

I'm trying to understand a(11) = 12 and a(12) = 30

Please explain exactly how you determine the value of a(n).

Alexander Povolotsky wrote:
> I did not calculate/check original terms (just used what was given).
> I am deferring to Paolo and Farideh to settle their terms differences ;-)
>
> Sory for previous copy and paste errors in typing array a(n) - I have
> difficulties with that ;-) .
> In my previous email  I used Paolo's results for calculating b(n) = a(n) -n
> Below is my another attempt for b(n) = a(n) -n for Farideh's variant
>
> a(n)=[2,1,4,3,6,5,8,7,10,9,12,30,14,20,16,22,18,23,21,31,33,34,40,25,
> 36,27,35,29,37,41,42,38,44,50,46,45,48,47,43,51,52,53,55,56,60,
> 57,58,59,54,61,62,63,64,66,67,70,68,69,71,72,73
> ,74,75,77,76,78,80,79,81,82,83,84,85,86,88,89,90,91,100,92,93,94,95,96,97,99,
> 98,101,102,111,200,103,104,105,106,107,108,110,112,222]
>   



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