Questions regarding divisibility in general are often a lot easier to tackle using modulaic algebra since we don't care about precisely what values the numbers will assume. In case you don't know what the modulo k operator is, it is basically the remainder that you get after you divide a number by k. If a number is divisible by k, so in our case 4, then it is zero in mod 4 since there is no rest.
So in this case we can translate the first values that you got into mod 4 as follows: 1,1,2,3,3,2 ... Now let's multiply a number in mod 2 by a number in mod 3. These numbers can be written as a=(4c+2) and b=(4d+3), so the result of their multiplication is 4d+6. However, we should add an extra one in the sequence, which will give us 4d+7. As you can verify, this number is indeed 3 in mod 4. Whenever we multiply two numbers that are 3 in mod 4, the next number in the sequence will be 2.
Therefore we can conjecture that no element in the entire sequence will be divisible by 4 since each one of them will be either 2 or 3 in mod 4.