Additive Persistence

Consider the process of taking a number, adding its Digits, then adding the Digits of 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 $n$ is called the additive persistence of $n$, and the Digit obtained is called the Digital Root of $n$.

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, ... (Sloane's A031286). The smallest numbers of additive persistence $n$ for $n=0$, 1, ... are 0, 10, 19, 199, 19999999999999999999999, ... (Sloane's A006050).

See also Digitaddition, Digital Root, Multiplicative Persistence, Narcissistic Number, Recurring Digital Invariant

References

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

Sloane, N. J. A. Sequences A031286 and A006050/M4683 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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