Along the path of the great Kaprekar: A-function, repunits and their properties

Following the famous Indian mathematician D. Kaprekar, in the paper5, the author presented a new method for obtaining integers ????(????) = ⅑ (???? − ????(????)) where S(n) is the sum of digits of the number n In decimal notation. This A-function turned out to be related to a remarkable class of numbers the class of integers Rn repunit. In the paper, new properties are found Rn. The properties A-function. It is proved that the set of a-self numbers is infinite and each a-self number has exactly 10 generators. The hypothesis about the distribution of a-self numbers is justified and formulated. The hypothesis about the distribution of the number of chains of «neighboring» a-self numbers is justified and formulated and the complete consistency of the two hypotheses is proved. Formulas for the chains of «neighboring» a-self numbers are found. The multiple relations between the Kaprekar function K(n) and the function introduced by us A(n). We studied and found all solutions of the functional equation A(qn) = qA(n).