Note that if n >= 16 then n# > n:
according to Chebyshev theorem there is a prime p between [n/2] and n, and there is a prime q between [n/4] and [n/2]. Therefore,
#n >= pq > [n/4]^2 >= sqrt(n)^2 = n.
Since
3# = 2*3 = 6
3## = 2*3*5 = 30 which is >= 16
from this point the sequence must be strictly increasing:
3### > 3##
3#### > 3###
and so on.
