[seqfan] If 2 disappears

[Ali Sada]

Hi Everyone, 

Because of a programming mistake, 2 disappeared. Subsequently, the prime numbers sequence changed. 
4,8, and 2p became “primes.” 
The new “prime numbers” sequence is:
3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, . . .  (A232803)

Now, the program tries to reconstruct the natural numbers using their prime factors:
a(1) = 1
a(2) = p1
a(3) = p2
a(4) = p1*p1
a(5) = p3
a(6) = p1*p2
etc.

Because of the mistake, the “natural numbers” sequence became:
1, 3, 4, 9, 5, 12, 6, 27, 16, 15, 7, 36, 8, 18, 20, 81, 10, 48, 11, 45, 24, 21, 13, 108, 25, 24, 64, 54, 14, 60, 17, 243, 28, . . .
(Obviously, 2 disappeared as a number, not as a digit.)
Because we have more primes, there will be more natural numbers, i.e. some numbers will appear more than once. What are these number? 

I would really appreciate it if you tell me this is a suitable sequence for the OEIS. 

Best,

Ali