Self Number

A self number is a positive integer (usually in base 10 unless specified otherwise) which has no digitaddition generator. Such numbers are also called Colombian numbers (Recamán and Bange 1974). There are infinitely many such numbers, since an infinite sequence of self numbers can be generated from the recurrence relation

(1)

for , 3, ..., where . The first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, ... (OEIS A003052).

An infinite number of 2-self numbers (i.e., base-2 self numbers) can be generated by the sequence

(2)

for , 3, ..., where and is the number of binary digits in . For an integer base , an infinite number of -self numbers can be generated from the sequence

(3)

for , 3, ..., and

(4)

Joshi (1973) proved that if the integer base is odd, then a positive integer is a -self number iff is odd. Patel (1991) proved that , , and are -self numbers in every even integer base .