Persistence of a number assumed. The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root. Put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root. The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4. Also, 39 is the smallest
Keith number appears in the sequence thus constructed. As an example, consider the 3-digit number "N" = 197. The sequence goes like this: Because 197 appears in the sequence, 197 is seen to be indeed a Keith number. A Keith number is a positive integer "N" that appears as a term in a linear recurrence relation with initial terms based on its own decimal digits. Given an "n"-digit number a sequence formula_2 is formed with initial terms formula_3 and with a general term produced as the sum of the previous "n" terms. If the number "N" appears in the sequence formula_2, then