[1] 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 the permutated sums: If zero is treated as a full digit in all positions, then 207 in base ten is a maximal Osiris number, being equal to the sum of all possible distinct numbers formed from permutated sub-samples of its digits: In other bases, maximal Osiris numbers exist that do not contain zeros.
These simplified tests considerably reduce the task of finding large Osiris numbers in a particular base.
For example, to test by brute force whether permutated six-digit samples of n = 332,639,667,360 are equal to n would involve summing 665,280 numbers, where 665,280 = 12 x 11 x 10 x 9 x 8 x 7 = 12!
However, because each digit of n occurs 55440 times in each of the six possible positions in the samples, the test is reduced to this: