User:Georg Fischer/Shared digits in squares

In 2009, Jonathan Wellons defined 336 OEIS sequences in the range A136808 - A137146. They are based on the decimal digits, but more interesting as it might seem at first sight. Some are perhaps finite, some seem to be hard. For some patterns a proof is outstanding (c.f. below). I think that some could have a connection to cellular automata.

Broken Link

Jonathan Wellons homepage jonathanwellons.com is no longer accessible. I wrote an email to the address cited on his archived pages, but without reply so far. In the meantime, I

• recovered the page (which he had linked in all OEIS sequences) from the Internet archive,
• replaced his internal links by those to the OEIS sequences,
• appended a numerical list of his OEIS sequences,
• stored the modified page temporarily under http://www.teherba.org/fasces/oeis/a136809.html.

In the LINKS section I would like to write:

J. Wellons, <a href="http://jonathanwellons.com/shared-digits/">Tables of Shared Digits</a> (broken link)
J. Wellons, <a href="/A136809/a136809.html">Tables of Digits</a> - recovered page with additional information by Georg Fischer

I tried to upload the modified HTML file, but it is automatically stored as

/A136809/a136809.html.txt

by the OEIS server. I would also like to incorporate parts of the following on that page.

Proofs

For obvious patterns in the sequences there were attempts to prove them, for example in :

A136859 Numbers n such that n and the square of n use only the digits 0, 1, 4 and 6.
0, 1, 4, 10, 40, 100, 400, 1000, 4000, 10000, 40000, 100000, 400000, 1000000, 
4000000, 10000000, 40000000, 100000000, 400000000, 1000000000, 4000000000, 
10000000000, 40000000000, 100000000000, 400000000000

I think that all patterns were up to 4 digits are followed by zeroes can be proved by the following arguments (I stick to the example for 0,1,4,6):

• a^2 mod b = ((a mod b) * (a mod b)) mod b. Therefore we only need to observe the m rightmost digits.
• The trailing zeroes are of no concern. If the digit 0 is allowed by the sequence, and if a(n) = d₁d₂ ... d_m with d_m > 0 is in the sequence, then a(n) * 10^k for any k > 0 is in the sequence.
• We square all possible 2-digit combinations without trailing 0 (01, 04, 06, 11, 14, 16, 41, 44, 46, 61, 64, 66). Only very few of these squares end with 2 of the allowed digits, namely:

 n    n^2
01   0001
04   0016
10   0100 - of no concern
40   1600 - of no concern
46   2116 (c.f. below)
60   3600 - of no concern

As Andrew pointed out, the rest of the proof is not complete for the 46.