Autobiographical numbers revisited

[Eric Angelini]

Hello Maximilian,

> - why just stop at 11 ??

... because this is how A046043 works !

A046043 starts with array of length 1 (no integer)
                               then 2 (no integer)
                               then 3 (no integer)
                               then 4 (2 integers, 1210, 2020)
                               then 5 (1 integer, 21200)
                               etc.

> go to other bases

... ah yes, good idea.

Best,
É.




-----Message d'origine-----
De : Maximilian Hasler [mailto:maximilian.hasler at gmail.com] 
Envoyé : mardi 17 juin 2008 16:06
À : Eric Angelini
Cc : SeqFan
Objet : Re: Autobiographical numbers revisited

>  Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |
> Occurrences | 5 | 3 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 2 | --> 53110100002
> More such numbers:
>  Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |
> Occurrences | 5 | 4 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 2 | 1 | --> 541011000021

this seems not very natural to me, since to be consistent, you should
then include all possible substrings, e.g. also "54", "41", "21" etc.
(at least up to a given length, say 2 digits) - why just stop at 11 ??
And also, you silently ignore the substrings "01", ... so in fact you
should not talk about "strings" since you consider numbers, not
strings.

A better justified alternative would be, IMHO, to go to other bases
(e.g. count hexadecimal digits using hexadecimal digits, etc.)
One maybe "interesting" non-base sequence would be:
a(n)=number of autobiographical numbers [in the original strict sense] in base n

Maximilian
PS:
A047841 should have kw "full" (I suppose, the b-file justifies this)
A138480 - the 3 additional terms could have been just a remark in
A046043, or vice versa ("3 last terms not allowed if "digit 10 etc" is
considered as nonsense")....