“Self Number”的概念、定义、习题解题方法及翻译-数学辞典

单词

Self Number

释义

Self Number

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

(1)

for

, 3, ..., where

. 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

(2)

for

, 2, ..., where

and

is the number of digits in

. An infinite number of

-selfnumbers can be generated from the sequence

(3)

for

, 3, ..., and

(4)

Joshi (1973) proved that if

is Odd, then

is a

-self number Iff

is Odd. Patel (1991) proved that

, and

are

-self numbers in every Even base

See also Digitaddition

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. SequenceA003052/M2404in ``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.

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