Digital Root

Consider the process of taking a number, adding its Digits, then adding the Digits of numbers 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 digital roots of the first few integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 9, 1, ... (Sloane's A010888). The digital root of an Integer $n$ can therefore be computed without actually performing the iteration using the simple congruence formula

$$\cases{ n {\rm\ (mod\ 9)} & $n$\ $\not\equiv$\ 0 (mod 9) \cr 9 & $n\equiv 0\ \left({{\rm mod\ } {9}}\right)$.\cr}$$

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

References

Sloane, N. J. A. Sequences A010888 and A007612/M1114 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.