Self prime

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 < 15 give a result < 20; all other n give a result > 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.

The first few base 10 self numbers are:

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.

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.

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.

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

In October 2006 Luke Pebody demonstrated that the largest known Mersenne prime that is at the same time a self number is 2^24036583−1. This is then the largest known self prime as of 2006^[update].

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