A number (usually base 10 unless specified otherwise) which has no Generator. Such
numbers were originally called Columbian Numbers (S. 1974). There are infinitely many such
numbers, since an infinite sequence of self numbers can be generated from the
Recurrence Relation

(1) |

An infinite number of 2-self numbers (i.e., base-2 self numbers) can be generated by the sequence

(2) |

(3) |

(4) |

**References**

Cai, T. ``On -Self Numbers and Universal Generated Numbers.'' *Fib. Quart.* **34**, 144-146, 1996.

Gardner, M. *Time Travel and Other Mathematical Bewilderments.* New York: W. H. Freeman, pp. 115-117, 122, 1988.

Joshi, V. S. Ph.D. dissertation. Gujarat University, Ahmadabad, 1973.

Kaprekar, D. R. *The Mathematics of New Self-Numbers.* Devaiali, pp. 19-20, 1963.

Patel, R. B. ``Some Tests for -Self Numbers.'' *Math. Student* **56**, 206-210, 1991.

S., B. R. ``Solution to Problem E 2048.'' *Amer. Math. Monthly* **81**, 407, 1974.

Sloane, N. J. A. Sequence
A003052/M2404
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-26