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 If these potential digit-sums are used in the formula 2 x digit-sum x 11, the digit-sum of the result will determine whether or not the result is an Osiris number. In 35964, each digit occurs 12 times in the ones, tens and hundreds position of the sums: The test for further five-digit Osiris numbers of the same form (sampling three digits) will use potential digit-sums between 15 = 1+2+3+4+5 and 35 = 5+6+7+8+9. When this range of digit-sums is tested, only 35964 returns the same digit-sum as that used in the formula. These simplified tests considerably reduce the task of