Additive Persistence

Consider the process of taking a number, adding its digits, then adding the digits of the number derived from it, etc., until the remaining number has only one digit. The number of additions required to obtain a single digit from a number  is called the additive persistence of , and the digit obtained is called the digital root of .

For example, the sequence obtained from the starting number 9876 is (9876, 30, 3), so 9876 has an additive persistence of 2 and a digital root of 3. The additive persistences of the first few positive integers are 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, ... (OEIS A031286). The smallest numbers of additive persistence  for , 1, ... are 0, 10, 19, 199, 19999999999999999999999, ... (OEIS A006050).

REFERENCES:

Hinden, H. J. "The Additive Persistence of a Number." J. Recr. Math. 7, 134-135, 1974.

Sloane, N. J. A. Sequences A006050/M4683 and A031286 in "The On-Line Encyclopedia of Integer Sequences."

Sloane, N. J. A. "The Persistence of a Number." J. Recr. Math. 6, 97-98, 1973.