Self Number

A number (usually base 10 unless specified otherwise) which has no digitaddition generator. Such numbers were originally called Colombian numbers (S. 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 , 2, ..., where and is the number of digits in . An infinite number of -self numbers can be generated from the sequence

(3)

for , 3, ..., and

(4)

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