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05AB1E, 20 bytes

Lpvyi1LÒN>¢àN>*ˆ}} ̄P

Explanation

Lpv # for each item in isprime(range(1,N)): N=7 -> [0,1,1,0,1,0,1]
 yi # if prime
 1LÒN>¢ # count occurrences of the prime 
 in the prime-factorization of range(1,N):
 p=2 -> [0,1,0,2,0,1,0]
 àN>*ˆ # add max occurrence of that prime multiplied by the prime 
 to global array: N=7 -> [4,3,5,7]
 }} # end if/loop
 ̄P # get product of global array

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