Self number
In number theory, a self 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, because no such combination can be found. 21 is not, because it can be written as 15 + 1 + 5 using n = 15. These numbers were first described in 1959 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|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.
Self numbers in specific bases
For base 2 self numbers, see.The first few base 10 self numbers are:
Self primes
A self prime is a self number that is prime.The first few self primes in base 10 are