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
Digit-reassembly number Digit-reassembly number Digit-reassembly numbers, or Osiris numbers, are numbers that are equal to the sum of permutations of sub-samples of their own digits (compare the dismemberment and reconstruction of the god Osiris in Egyptian mythology). For example, 132 = 12 + 21 + 13 + 31 + 23 + 32. In base ten, the smallest Osiris numbers are these, with a number-length of three digits and digit-span of two for the permutated sums: Note that all are multiples of 132. A larger Osiris number in base ten is this, with a number-length of five digits and digit-span of three for