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
Sum-product number Sum-product number A sum-product number is an integer that in a given base is equal to the sum of its digits times the product of its digits. Or, to put it algebraically, given an integer "n" that is "l" digits long in base "b" (with "d" representing the "x"th digit), if: formula_1 then "n" is a sum-product number in base "b". In base 10, the only sum-product numbers are 0, 1, 135, 144 . Thus, for example, 144 is a sum-product number because 1 + 4 + 4 = 9, and 1 × 4 × 4 = 16, and 9