[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: http://www.cetteadressecomportecinquantesignes.com/SubStrings.htm)