Fundamental Theorem of Arithmetic — ARC-style

Answer
For n = 202692987, the prime factors are 3 * 3 * 7 * 829 * 3881, the prime-power form is 3^2 * 7 * 829 * 3881, and the product of these factors is 202692987 with 4 distinct primes.

Reason Why
Existence in this run comes from repeated smallest-divisor decomposition: 202692987 factors as 3 * 3 * 7 * 829 * 3881. Each distinct factor is prime. For uniqueness up to order, the reverse traversal gives 3881 * 829 * 7 * 3 * 3, and both traversals sort to the same multiset of primes. So this concrete case exhibits the Fundamental Theorem of Arithmetic for 202692987: a prime factorization exists and is unique up to order. The extreme prime factors are 3 and 3881.

Check
C1 OK - repeated smallest-divisor decomposition produced the expected smallest-first factor list (3 3 7 829 3881).
C2 OK - the product of the computed factors reconstructs n = 202692987.
C3 OK - every distinct factor in the decomposition is prime by trial division.
C4 OK - the prime-power form matches 3^2 * 7 * 829 * 3881.
C5 OK - smallest-first and largest-first traversals have the same multiset of primes, so the factorization agrees up to order.
C6 OK - the smallest and largest prime factors are 3 and 3881.