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Prime factorization exponents form a geometric progression

Source: MEMO 2017 T8

August 25, 2017
number theorygeometric sequence

Problem Statement

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.
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