Primary N3 case: n = 202692987 has prime factors 3 * 3 * 7 * 829 * 3881.
primary prime-power form : 3^2 * 7 * 829 * 3881
sample count : 6
largest sample : 600851475143
total prime factors counted with multiplicity : 31
distinct primes seen across samples : 17
Sample factorizations:
360360 = 2^3 * 3^2 * 5 * 7 * 11 * 13
202692987 = 3^2 * 7 * 829 * 3881
4294967295 = 3 * 5 * 17 * 257 * 65537
600851475143 = 71 * 839 * 1471 * 6857
9876543210 = 2 * 3^2 * 5 * 17^2 * 379721
9999999967 = 9999999967
Existence comes from repeated smallest-divisor decomposition.
At each step, the first divisor found is prime because no smaller
positive divisor can divide the current number.
Smallest-divisor trace for the N3 source number:
202692987 = 3 * 67564329
67564329 = 3 * 22521443
22521443 = 7 * 3217349
3217349 = 829 * 3881
3881 is prime
Uniqueness up to order is checked by reversing each traversal and sorting
both factor lists. Matching sorted lists describe the same multiset of
prime factors, even when the factors were discovered in the opposite order.
source smallest-first factors : 3 * 3 * 7 * 829 * 3881
source largest-first factors : 3881 * 829 * 7 * 3 * 3
source sorted comparison : 3 * 3 * 7 * 829 * 3881
The additional samples cover repeated small factors, special products,
large composites, and a larger prime that has no smaller divisor.