In number theory, a self number or Devlali number in a given number base is a natural number that cannot be written as the sum of any other natural number and the individual digits of . 20 is a self number (in base 10), because no such combination can be found (all give a result less than 20; all other give a result greater than 20). 21 is not, because it can be written as 15 + 1 + 5 using n = 15. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar. Definition and properties Let be a natural number. We define the -self function for base to be the following: where is the number of digits in the number in base , and is the value of each digit of the number. A natural number is a -self number if the preimage of for is the empty set. 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 is infinite and has a positive asymptotic density: when is odd, this density is 1/2. Recurrent formula The following recurrence relation generates some base 10 self numbers: (with C1 = 9) And for binary numbers: (where j stands for the number of digits) we can generalize a recurrence relation to generate self numbers in any base b: 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. Selfness tests Reduction tests Luke Pebody showed (Oct 2006) that a link can be made between the self property of a large number n and a low-order portion of that number, adjusted for digit sums: Effective test Kaprekar demonstrated that: is self if Where: is the sum of all digits in . is the number of digits in . Self numbers in specific bases For base 2 self numbers, see . (written in base 10) The first few base 10 self numbers are: 1, 3, 5, 7, 9, 20, 31, 42,