Self number


In number theory, a self number, Colombian 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, 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 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:
And for binary numbers:
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 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:
Where:

Self numbers in specific bases b

For base 2 self numbers, see.
The first few base 10 self numbers are:
In base 12, the self numbers are:

Self primes

A self prime is a self number that is prime.
The first few self primes in base 10 are
The first few self primes in base 12 are:
In October 2006 Luke Pebody demonstrated that the largest known Mersenne prime in base 10 that is at the same time a self number is 224036583−1. This is then the largest known self prime in base 10.

Extension to negative integers

Self numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.

Excerpt from the table of bases where 2007 is self or Colombian

The following table was calculated in 2007.
BaseCertificateSum of digits
4048
41
4240
43
4436
4479
45
4681
47
48
49
5048
51
5260
53
5476
55
5641
57
5863
59
6089