# A conjecture

Richard Mathar mathar at strw.leidenuniv.nl
Tue Jun 17 11:50:46 CEST 2008

```
[a private mail from Neil gives me the opportunity to ask
the same unanswered questions I posted a few months ago]

----- Autobiographical numbers revisited -----

2020 is an autobiographical number because 2020 describes its
own "digit content" like this:

Digit:  0 | 1 | 2 | 3 |
Occurrences:  2 | 0 | 2 | 0 |

("In 2020 there are 2 zeros
0 one
2 twos
0 three")

This method gives the traditionnal (finite) list of autobio-
graphical numbers:

[http://www.research.att.com/~njas/sequences/A046043]

1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000.

Should we count substrings instead of digits, then we could
prolong the seq with a few new terms. Example:

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

Two "10" substrings: ---> ^^^^^   ^^^^^

Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |
Occurrences | 6 | 2 | 2 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | --> 62200010001

The "10" substring:  ------------->   ^^^^^

Sequence A046043 becomes:
1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 53110100002, 62200010001.

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
^^^^^^^^^
(one "10" substring and the "11" substring are interleaved in "110")

Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 |
Occurrences | 6 | 4 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 3 | 1 | 0 | --> 6401101000310

Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 |13 |
Occurrences | 7 | 4 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 3 | 1 | 0 | 0 | --> 74011001003100

Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 |13 |14 |
Occurrences | 8 | 4 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 3 | 1 | 0 | 0 | 0 | --> 840110001031000

Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 |13 |14 |15 |
Occurrences | 9 | 3 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | 0 | 1 | 0 | 0 | 0 | --> 9321000001201000

Sequence A046043 becomes:
1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 53110100002, 62200010001, 541011000021, 6401101000310, 74011001003100, 840110001031000, 9321000001201000.

Can one go further? I guess not. Let's explain why with
this example:

Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 |13 |14 |15 |16 |17 |
Occurrences  11   6   0   0   1   0   1   0   0   0   4   1   0   0   0   0   1   0

The array says the truth -- but how can one read the
number N it produces?

This number N is: 1160010100041000010.

With the above adopted rule the first "1" of N means
that "there is only 1 zero in N" -- which is false.

How could the reader know that he has to link the first
two "1"'s of N -- only method saying the truth about N:
"there are 11 one's in N".

Is all this worth a new entry in the OEIS or a new "comment"
in A046043? Is this way of counting substrings well-defined?
If yes, are there numbers (having less than 15 digits) which
were forgotten?

Best,
É.

(the same in french:

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