Additive Persistence

Consider the process of taking a number, adding its Digits, then adding the Digits ofnumber derived from it, etc., until the remaining number has only one Digit. The number of additions required toobtain a single Digit from a number is called the additive persistence of , and the Digit obtained iscalled 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 anda 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 for , 1, ... are0, 10, 19, 199, 19999999999999999999999, ... (Sloane's A006050).

References

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

Sloane, N. J. A. SequencesA031286 andA006050/M4683in ``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.

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