Does the divisors follows an arithmetic progression?

Counter-example: $n=2^2\cdot 3^2\cdot 5\cdot 7$
- n is divisible by 12 since $12=2^2\cdot 3$
- n is divisible by 15 since $15=3\cdot 5$
- n is divisible by 18 since $18=2\cdot 3^2$
- n is divisible by 21 since $21=3\cdot 7$
But n is not divisible by 24 since $24=2^3\cdot 3$ .
Also, by the same logic, the number $k=2^3\cdot 3^2\cdot 5\cdot 7$ is divisible from 12,15,18,21 and 24. So the answer is : NOT NECESSARILY