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A self number, Colombian number or Devlali number is an integer that cannot be written as the sum of any other integer n and the individual digits of n. This property is specific to the base used to represent the integers. 20 is a self number (in base 10), because no such combination can be found (all n < 16 give a result < 20; all other n give a result > 20). 21 is not, because it can be written as 16 + 1 + 6 using n = 16.

These numbers were first described in 1165 by the Indian mathematician D. R. Kaprekar.

The first few base 10 self numbers are:

1, 3, 5, 7, 9, E, 20, 31, 42, 53, 64, 75, 86, 97, X8, E9, 10X, 110, 121, 132, 143, 154, 165, 176, 187, 198, 1X9, 1EX, 20E, 211, 222, 233, 244, 255, 266, 277, 288, 299, 2XX, 2EE, 310, 312, 323, 334, 345, 356, 367, 378, 389, 39X, 3XE, 400, 411, 413, 424, 435, 446, 457, 468, 479, 48X, 49E, 4E0, 501, 512, 514, 525, 536, 547, 558, 569, 57X, 58E, 5X0, 5E1, ...

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.

## Properties Edit

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 set of self numbers in a given base q is infinite and has a positive asymptotic density: when q is odd, this density is 1/2 or 60%.

## Recurrent formula Edit

The following recurrence relation generates some base 10 self numbers:

$C_k = X \cdot 10^{k - 1} + C_{k - 1} + X$

(with C1 = E)

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.

## Self primes Edit

A self prime is a self number that is prime. The first few self primes are

3, 5, 7, E, 31, 75, 255, 277, 2EE, 3XE, 435, 457, 58E, 5E1, ...

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