|
|
A046043
|
|
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.
|
|
13
|
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
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.
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
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.
Enumerated by David Castro (david_castro(AT)retek.com).
|
|
REFERENCES
|
E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
M. Gardner, Mathematical Circus, pp. 128; 135 Prob. 7 Alfred A. Knopf NY 1979.
Tanya Khovanova, A Story of Storytelling Numbers, Math. Horizons, Sep 2009, 14-17.
|
|
LINKS
|
|
|
EXAMPLE
|
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".
|
|
MATHEMATICA
|
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 *)
|
|
CROSSREFS
|
Compare with the "Look-and-Say" version A047841.
|
|
KEYWORD
|
nonn,base,nice,fini,full
|
|
AUTHOR
|
Robert Leduc (leduc(AT)macalester.edu)
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|