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.

1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000

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

Table of n, a(n) for n=1..7.

Fred Gavin, Letters to the editor, Math. Mag 66 (4) (1993) p 276.

Amy Harmon, Beyond 'Hidden Figures': Nurturing New Black and Latino Math Whizzes, New York Times, Feb 17, 2017.

Tanya Khovanova, Autobiographical Numbers, arXiv:0803.0270 [math.CO], 2008.

Chris Smith, #MegaFavNumbers Self-Descriptive Numbers (the beauty 6210001000), video (2020)

K. Uhland, The Ten-Digit Number [Broken link?]

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

Cf. A138480 (another version), A359049.

Compare with the "Look-and-Say" version A047841.

Sequence in context: A186864 A338963 A135239 * A358538 A359049 A358711

Adjacent sequences: A046040 A046041 A046042 * A046044 A046045 A046046

Keyword

nonn,base,nice,fini,full

Author

Robert Leduc (leduc(AT)macalester.edu)

Extensions

Edited by N. J. A. Sloane, Aug 15 2019

Status

approved