Self Number

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

$$C_k=8\cdot 10^{k-1}+C_{k-1}+8,$$ (1)

for $k=2$, 3, ..., where $C_1=9$. The first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, ... (Sloane's A003052).

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

$$C_k=2^j+C_{k-1}+1$$ (2)

for $k=1$, 2, ..., where $C_1=1$ and $j$ is the number of digits in $C_{k-1}$. An infinite number of $n$-self numbers can be generated from the sequence

$$C_k=(n-2)n^{k-1}+C_{k-1}+(n-2)$$ (3)

for $k=2$, 3, ..., and

$$C_1=\cases{ n-1 & for $n$\ even\cr n-2 & for $n$\ odd.\cr}$$ (4)

Joshi (1973) proved that if $k$ is Odd, then $m$ is a $k$-self number Iff $m$ is Odd. Patel (1991) proved that $2k$, $4k+2$, and $k^2+2k+1$ are $k$-self numbers in every Even base $k\geq 4$.

See also Digitaddition

References

Cai, T. ``On $k$-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 $k$-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.