self number

An integer $n$ that in a given base $b$ lacks a digitaddition generator.

Consider, for example, the integer 41 in base 10. It can be expressed as 34 + 3 + 4. For 42, however, there is no such digitaddition, hence it is a self number.

If $2|b$, all odd will be self numbers.

Though self numbers form a small proportion of most ranges of $2b$ consecutive integers, there are infinitely many of them: The recurrence relation $S_i=(b-2)b^{i-1}+S_{i-1}+(b-2)$ (with $S_1=b-1$ if $2|b$ and $S_1=b-2$ otherwise) will give an incomplete though infinite list of self numbers.

Reference

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

Title	self number
Canonical name	SelfNumber
Date of creation	2013-03-22 15:56:15
Last modified on	2013-03-22 15:56:15
Owner	CompositeFan (12809)
Last modified by	CompositeFan (12809)
Numerical id	6
Author	CompositeFan (12809)
Entry type	Definition
Classification	msc 11A63
Synonym	Columbian number
Synonym	Colombian number
Synonym	self-number