%I #40 Jan 20 2023 08:03:21
%S 1210,2020,21200,3211000,42101000,521001000,6210001000
%N Autobiographical numbers (or curious numbers): list of numbers m = x_0 x_1 x_2 ... x_{b-1} (written in base b) such that x_i is the number of "digits" in m that are equal to i, for all i=0,...,b-1.
%C Note that the base b is the total number of "digits" in m. Since the numbers are written without spaces between the digits x_i, we must take b <= 10.
%C There are no such numbers for b<=3 or b=6, two such numbers for b=4, and exactly one such number for b=5 and each b>=7. - _David Callan_, Feb 17 2017
%C The proof of completeness is based on: x_0 > 0; x_i > 2 only if i = 0; for i > 2, x_i = 1 if i = x_0, x_i = 0 otherwise.
%C Enumerated by David Castro (david_castro(AT)retek.com).
%D E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
%D M. Gardner, Mathematical Circus, pp. 128; 135 Prob. 7 Alfred A. Knopf NY 1979.
%D Tanya Khovanova, A Story of Storytelling Numbers, Math. Horizons, Sep 2009, 14-17.
%H Fred Gavin, <a href="http://www.jstor.org/stable/2690749">Letters to the editor</a>, Math. Mag 66 (4) (1993) p 276.
%H Amy Harmon, <a href="https://www.nytimes.com/2017/02/17/nyregion/new-york-math-camp.html">Beyond 'Hidden Figures': Nurturing New Black and Latino Math Whizzes</a>, New York Times, Feb 17, 2017.
%H Tanya Khovanova, <a href="http://arxiv.org/abs/0803.0270">Autobiographical Numbers</a>, arXiv:0803.0270 [math.CO], 2008.
%H Chris Smith, <a href="https://www.youtube.com/watch?v=nlSdg3JOXcs">#MegaFavNumbers Self-Descriptive Numbers (the beauty 6210001000)</a>, video (2020)
%H K. Uhland, <a href="http://uhlandkf.homestead.com/files/PuzzlePage/199904Sol.htm">The Ten-Digit Number</a> [Broken link?]
%e m = 1210 is written in base 4 (since it has 4 digits), and has one 0, two 1's, one 2 and zero 3's and m = "one two one zero".
%t isSelfDescribing[n_Integer] := (RotateRight[DigitCount[n]] == PadRight[IntegerDigits[n], 10]); Select[Range[10^10 - 1], isSelfDescribing] (* _Martin Ettl_, Oct 09 2012 *) (* Warning: This program causes Mathematica to crash! - _David Callan_, Feb 17 2017 *)
%Y Cf. A138480 (another version), A359049.
%Y Compare with the "Look-and-Say" version A047841.
%K nonn,base,nice,fini,full
%O 1,1
%A Robert Leduc (leduc(AT)macalester.edu)
%E Edited by _N. J. A. Sloane_, Aug 15 2019