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 b-1, where b is the base of the sequence. This sequence is in base b=10.

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

Offset: 1

Author

Marc Morgenegg, Nov 21 2022

Keywords

• nonn
• fini
• base

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^((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

Examples

			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.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Michael S. Branicky, Python program

Crossrefs

Cf. A046043, A138480.

Programs

Python

# see linked program

Extensions

a(14)-a(19) inserted and a(25) from Michael S. Branicky, Nov 21 2022