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COMMENTS
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The composition length of a finite solvable group is equal to the number of prime factors of the order, counting multiplicities. Thus, for example, the symmetric group of permutations on 4 elements is a solvable group of derived length 3, and its order is 24=2*2*2*3 which has 4 prime divisors. This is the smallest possible number of factors for a solvable group of derived length 3, so a(3) = 4.
The sequences is monotonic increasing, and the difference cannot be 1 twice in a row. Thus the smallest possible differences are 1,2,1,2.., which is what we see for the first 5 differences. The sixth difference is 5 which breaks that pattern. Glasby shows that a(n) grows exponentially, with exponent at least 1.3, so in the long run the differences must often be very large. However, it seems that the difference 1 may show up infinitely often.
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