Primorial question

[Dougy]

For the question I need a slightly modified definition of primorial. Let n# = the product of all primes less than or equal to n, for all natural numbers n. Note that this disobeys the convention that n must be prime for n# to be valid.

Does the sequence 3#, 3##, 3###, ... get arbitarily large? Or could there exist a (non-trivial) natural number k such that k#=k##?

[maxal]

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.

[Dougy]

Thanks.