A self number or Columbian number is an integer which, in a given base, cannot be generated by any other integer added to the sum of its digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is, 21 = 15 + 1 + 5. No such sum will generate the integer 20, hence it is a self number. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar.

The first few base 10 self numbers are

1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, 501, 512, 514, 525

In general, for even bases, all odd numbers below the base number are self numbers, since any number below such an odd number would have to also be a 1-digit number which when added to its digit would result in an even number. For odd bases, all odd numbers are self numbers.

The following recurrence relation generates base 10 self numbers:

$C\_k\; =\; 8\; \backslash cdot\; 10^\{k\; -\; 1\}\; +\; C\_\{k\; -\; 1\}\; +\; 8$

(with C₁ = 9)

And for binary numbers:

$C\_k\; =\; 2^j\; +\; C\_\{k\; -\; 1\}\; +\; 1$

(where j stands for the number of digits) we can generalize a recurrence relation to generate self numbers in any base b:

$C\_k\; =\; (b\; -\; 2)b^\{k\; -\; 1\}\; +\; C\_\{k\; -\; 1\}\; +\; (b\; -\; 2)$

in which C₁ = b − 1 for even bases and C₁ = b − 2 for odd bases.

The existence of these recurrence relations shows that for any base there are infinitely many self numbers.

A search for self numbers can turn up self-descriptive numbers, which are similar to self numbers in being base-dependent, but quite different in definition and much fewer in frequency.

See Also

Self Prime

Base-dependent integer sequences

Auto nombre | Numero colombiano | Samoštevilo | 自我数