| Self-descriptive Number |
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| base-dependent integer sequences | |
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For example, in base 10, the number 6210001000 is self-descriptive because it has six 0s, two 1s, one 2, one 6, and no 3s, 4s, 5s, 7s, 8s or 9s. There are no self-descriptive numbers in bases 2, 3 or 6. In bases 7 and above, there is, if nothing else, a self-descriptive number of the form , which has ''b'' - 4 instances of the digit 0, two instances of the digit 1, one instance of the digit 2, one instance of digit ''b'' - 4, and no instances of any other digits. The following table lists some self-descriptive numbers in a few selected bases: Sloane's lists a few more self-descriptive numbers. From the numbers listed in the table, it would seem that all self-descriptive numbers have digit sums equal to their base, and that they're multiples of that base. That a self-descriptive number in base ''b'' must be a multiple of that base can be proven ad absurda as follows: assume that there is in fact a self-descriptive number ''m'' in base ''b'' that is ''b''-digits long but not a multiple of ''b''. The digit at position ''b'' - 1 must be at least 1, meaning that there is at least one instance of the digit ''b'' - 1 in ''m''. At whatever position ''x'' that digit ''b'' - 1 falls, there must be at least ''b'' - 1 instances of digit ''x'' in ''m''. Therefore, we have at least one instance of the digit 1, and ''b'' - 1 instances of ''x''. If ''x'' > 1, then ''m'' has more than ''b'' digits, leading to a contradiction of our initial statement. And if ''x'' = 0 or 1, that also leads to a contradiction. The concept of self-descriptive numbers is similar to that of autobiographical or curious numbers, except that there is no digit length requirement for autobiographical numbers. (Sloane's lists base 10 autobiographical numbers). Self-descriptive numbers are like Self Number s only in that they're both base-dependent concepts. EXTERNAL REFERENCES
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