

A358538


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 b1, where b is the base of the sequence. This sequence is in base b=10.


1



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
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OFFSET

1,1


COMMENTS

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^((b1)^2  2)  (b1)*b^((b1)*b  1) + ((b^(b1)  1)/(b1)) * b^(b1) * ((b1)*b^(b*(b1))  b^((b1)^2 + 1) + 1)/(b^(b1)  1)^2. The baseb expansion of this number is the concatenation of b2 digits b1, 1 digit b2, 1 digit b1, b3 digits b2, and b1 digits k for each k in b3..0. This is equivalent to taking a string of digits consisting of b1 copies of every valid baseb digit (0..b1), sorting its digits in descending order, removing one of the digits b2, and then swapping the positions of the last digit b1 and the first digit b2. (Thus, for b=10, the base10 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


LINKS



EXAMPLE

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 baseb expansion count the occurrences of the corresponding digit. In this case, the last digit, 1, is the 11th.


PROG

(Python) # see linked program


CROSSREFS



KEYWORD

nonn,fini,base


AUTHOR



EXTENSIONS



STATUS

approved



