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A358538
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Autobiographical numbers: the first digit of the term counts how many 0's the term in total has, the second how many 1's etc. up to the last digit but no more than b-1, where b is the base of the sequence. This sequence is in base b=10.
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1
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1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 52200100019, 52200100108, 52200101007, 52200110006, 52201100004, 52210100003, 53010100019, 53010100108, 53010101007, 53010110006, 53011100004, 53110100002, 61200020006, 62200010001, 63010010001, 70200002007, 72100001000, 431110000299
(list;
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refs;
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history;
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internal format)
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OFFSET
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1,1
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COMMENTS
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Other terms include: 640010100011, 722000010001, 722000010010, 730100010001, 730100010010, 802000002008, 802000002080, 821000001000, 6500011000111, 7400100100011, 7400100100101, 8301000010001, 9020000002009, 9020000002090, 9020000002900, 9210000001000.
This sequence is finite. The last term starts with 99999999898... and has 89 digits.
This sequence is for base b=10. For each base b > 2, the last term of the corresponding sequence has b^2 - b - 1 digits.
For b > 2, the final term of the sequence equals b^((b-1)^2 - 2) - (b-1)*b^((b-1)*b - 1) + ((b^(b-1) - 1)/(b-1)) * b^(b-1) * ((b-1)*b^(b*(b-1)) - b^((b-1)^2 + 1) + 1)/(b^(b-1) - 1)^2. The base-b expansion of this number is the concatenation of b-2 digits b-1, 1 digit b-2, 1 digit b-1, b-3 digits b-2, and b-1 digits k for each k in b-3..0. This is equivalent to taking a string of digits consisting of b-1 copies of every valid base-b digit (0..b-1), sorting its digits in descending order, removing one of the digits b-2, and then swapping the positions of the last digit b-1 and the first digit b-2. (Thus, for b=10, the base-10 expansion of the final term is the concatenation of eight 9's, one 8, one 9, seven 8's, nine 7's, nine 6's, ..., nine 1's, and nine 0's.) - Jon E. Schoenfield, Nov 21 2022
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LINKS
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EXAMPLE
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63010010001 is a term: we have six 0's, three 1's, one 3 and one 6 as digits in the term, visualized as follows:
Digits: 0123456789
term: 63010010001.
Note that this example also shows, starting from the 11th digit, there is no more representation of the frequency of that digit, because only the first b digits of its base-b expansion count the occurrences of the corresponding digit. In this case, the last digit, 1, is the 11th.
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PROG
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(Python) # see linked program
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CROSSREFS
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KEYWORD
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nonn,fini,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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