OEIS A212558: Proof of Unproven Conjecture? Proven!

[Paul Dale]

The same property seems to hold for other bases apart from 10:

Code:


BS  MAXn Native
 2   255 11111111
 3   485 122222
 4   767 23333
 5   624 4444
 6   295 5555
 7  2057 5666
 8  2047 3777
 9  2915 3888
10  2999 2999
11  3992 2aaa
12  3455 1bbb
13  4393 1ccc
14  5487 1ddd
15  6749 1eee
16  8191 1fff
17  4912 ggg
18  5831 hhh
19  6858 iii
20  7999 jjj
21  9260 kkk
22 10647 lll
23 12166 mmm
24 13823 nnn
25 15624 ooo
26 17575 ppp
27 19682 qqq
28 21951 rrr
29 24388 sss
30 26999 ttt
31 29790 uuu
32 32767 vvv
33 35936 www
34 39303 xxx
35 39199 vyy
36 38879 tzz

BS is the base, MAXn is the largest n for which a(n) is non-zero. Native is n presented in the specified base rather than base 10.

Some observations:

• The final digits are always a run of the highest digit permitted in the base.
• The binary bases apart from four (2, 8, 16, 32) are equivalent to a run of set bits in binary (no gaps). Why is base 4 special?
• The sequence is generally increasing but dips between bases 16 and 17.
  

It is interesting to note that the final number always ends with a run of the largest digit permitted by the base.