MathDB

8

Part of 2017 Middle European Mathematical Olympiad

Problems(1)

Prime factorization exponents form a geometric progression

Source: MEMO 2017 T8

8/25/2017
For an integer n≥3n \geq 3 we define the sequence α1,α2,…,αk\alpha_1, \alpha_2, \ldots, \alpha_k as the sequence of exponents in the prime factorization of n!=p1α1p2α2…pkαkn! = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}, where p1<p2<…<pkp_1 < p_2 < \ldots < p_k are primes. Determine all integers n≥3n \geq 3 for which α1,α2,…,αk\alpha_1, \alpha_2, \ldots, \alpha_k is a geometric progression.
number theorygeometric sequence