no because there are 3 numbers which are 2 mod 4 amongst 12 consecutive numbers. and so by the assumption they must all be of the form $2p,2q,2r$ for p,q,r prime. so we have: that the primes $p,q,r$ are each separated by 2. but in this case there is a contradiction modulo 3 unless the smallest is divisible by 3 and hence must be 3. so p=3 and so our list contains the numbers $6,8,10$. But in any case it will contain $7$ which does not satisfy the properties. so we conclude there's no such sequence of numbers