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
Happy number "happy base". The only known happy bases are 2 and 4. There are no others less than 500,000,000. Base 3 is also a special case in that the happiness (or sadness) of a number is an indication also of being odd (or Even). Specifically, because 3 - 1 = 2, the sum of every digit of a base 3 number will indicate divisibility by 2 IFF the sum of digits ends in 0 or 2. This is the general application of the test for 9-divisibility in base 10. In balanced ternary, the digits are 1, -1 and 0. The square